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∃:∀Z :Z ↔ φ Z ∃:∀Z :Z ↔ φ Z
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Z
φ ∀[ φ [ → φ [ → ∀Z 00 Z → φ Z
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HKTUV
∃:∀Z :Z ↔ φ Z
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(KPCNN[ VJGTG KU VJG OCVVGT QH KPHKPKV[ 5WRRQUG HQT C OQOGPV VJCV YG CFF VQ 4WUUGNNOU QTKIKPCN U[UVGO (TGIGOU CUUWORVKQP QH VJG GZKUVGPEG QH GZVGPUKQPU +V YKNN DG KPUVTWEVKXG VQ EQPUKFGT JQY VJG RTGFKECVKXKV[ TGUVTKEVKQPU UCXG WU HTQO RCTCFQZ 9G FQ KPFGGF JCXG VJG HQNNQYKPI KPUVCPEG QH EQORTGJGPUKQP
∃:∀Z :Z ↔ ∃; Z
U
;
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+H YG NGV DG UWEJ CP CU KP VJGP YJCV YG ECP EQPENWFG HTQO KU VJCV ECPPQV DG EQGZVGPUKXG YKVJ CP[ HKTUVTQWPF EQPEGRV (QT NGVVKPI DG CP[ HKTUVTQWPF EQPEGRV CPF KVU GZVGPUKQP VGNNU WU VJCV HCNNU WPFGT KH CPF QPN[ KH KV FQGU PQV HCNN WPFGT $WV VJKU EQPENWUKQP KU KP KVUGNH PQ EQPVTCFKEVKQP 6JKU EQPENWUKQP FQGU JQYGXGT EQPURKEWQWUN[ EQPVTCFKEV VJG CZKQO QH TGFWEKDKNKV[ 9G JCXG VJGTGHQTG VQ EJQQUG DGVYGGP VJCV CZKQO CPF VJG CZKQO QH VJG GZKUVGPEG QH GZVGPUKQPU 4WUUGNN OCFG JKU EJQKEG HQT VJG HQTOGT 9JCV YKNN DG GZRNQTGF KP VJG PGZV EJCRVGT KU YJCV JCRRGPU KH QPG EJQQUGU VJG NCVVGT 6JG CPUYGT VJCV YKNN GXGPVWCNN[ GOGTIG KU VJCV QYKPI VQ RTQDNGOU QXGT OCVJGOCVKECN KPFWEVKQP OGPVKQPGF GCTNKGT KP VJKU UGEVKQP QPG IGVU QPN[ C HTCIOGPV QH CTKVJOGVKE $WV CU C OCVVGT QH HCEV YKVJ 4WUUGNNOU EJQKEG QPG IGVU PQ CTKVJOGVKE CV CNN YKVJQWV C UGEQPF GZVTC CUUWORVKQP DG[QPF TGFWEKDKNKV[ 6Q DGIKP YKVJ VJGTG ECP DG PQ SWGUVKQP QH QDVCKPKPI PWODGTU CU M QT CU 4WUUGNN ECNNU VJG GPVKVKGU CV VJG DQVVQO QH VJG JKGTCTEJ[ QH NGXGNU M CPF HQT 4WUUGNN C PWODGT YKNN JCXG VQ DG PQV CU YKVJ (TGIG VJG GZVGPUKQP QH C UGEQPFNGXGN EQPEGRV DWV UKORN[ VJG UGEQPFNGXGN EQPEGRV KVUGNH 6JG PWODGT VYQ KU VJG UGEQPFNGXGN EQPEGRV QH DGKPI C HKTUVNGXGN EQPEGRV WPFGT YJKEJ VJGTG HCNNU CP QDLGEV CPQVJGT FKUVKPEV QDLGEV DWV PQ HWTVJGT QDLGEV FKUVKPEV HTQO DQVJ QH VJGUG 6JKU KU YJ[ CU OGPVKQPGF GCTNKGT 4WUUGNN JCU VQ CUEGPF JKIJGT KP VJG JKGTCTEJ[ QH NGXGNU VJCP (TGIG GXGT FKF GURGEKCNN[ YJGP JG OQXGU QP VQ VJG VJGQT[ QH TGCN PWODGTU 9KVJQWV VJG CUUWORVKQP QH GZVGPUKQPU VQ IKXG WU QNQIKECN QDLGEVUR KV YKNN DG KORQUUKDNG VQ RTQXG VJG 2GCPQ RQUVWNCVGU HQT PWODGTU UQ EQPEGKXGF 6JG RTQDNGO KU PQV QXGT KPFWEVKQP DWV QXGT VJG 2GCPQ RQUVWNCVGU ,WUV YJKEJ QPG QH VJGO YKNN DG WPRTQXCDNG FGRGPFU QP LWUV JQY VJG FGHKPKVKQP QH UWEEGUUQT KU HTCOGF DWV VJG[ EGTVCKPN[ ECPPQV DG RTQXGF DGECWUG VQIGVJGT VJG[ IWCTCPVGG VJG GZKUVGPEG QH KPHKPKVGN[ OCP[ PWODGTU CPF YKNN DG KORQUUKDNG VQ RTQXG (QT YKVJQWV QNQIKECN QDLGEVUR KV YKNN DG KORQUUKDNG HQT NQIKE VQ RTQXG VJG GZKUVGPEG QH KPHKPKVGN[ OCP[ M QT HQT VJCV OCVVGT QH M QDLGEVU CPF KH VJGTG CTG QPN[ QDLGEVU VJGTG YKNN DG CV OQUV P HKTUVNGXGN EQPEGRVU #
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: 4WUUGNNHKPKVG :
CZKQO QH EJQKEG
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#U OCVJGOCVKEKCPU VQQM VJCV EQWTUG (TGIGOU U[UVGO YCU KPETGCUKPIN[ HQTIQVVGP WPVKN VJG TGEGPV PGQ(TGIGCP TGXKXCN 'PQWIJ PQVKQPU JCXG DGGP KPVTQFWEGF KP VJG UWOOCT[ QH (TGIGOU CPF 4WUUGNNOU CRRTQCEJGU VQ OCMG KV RQUUKDNG CV VJKU RQKPV VQ UMGVEJ VJG FKTGEVKQPU KP YJKEJ PGQ(TGIGCPU JCXG RTQEGGFGF +H QPG NQQMU ENQUGN[ CV VJG EQPVTCFKEVKQP KP (TGIGOU U[UVGO KV YKNN DG UGGP VQ JCXG HGCVWTGU QH YJKEJ EQPVTKDWVGU VQ VJG RCTCFQZ (KTUV (TGIG KU XGT[ HTGG KP CUUWOKPI VJG GZKUVGPEG QH EQPEGRVU CPF KP RCTVKEWNCT JKU U[UVGO CNNQYU VJG KPVTQFWEVKQP QH VJG n ∉ 5GEQPF (TGIG KU XGT[ HTGG KP CUUWOKPI VJG GZKUVGPEG QH GZVGPUKQPU HQT EQPEGRVU CPF KP RCTVKEWNCT JKU U[UVGO CNNQYU VJG KPVTQFWEVKQP QH VJG ] ∉ _ 0GQ(TGIGCP VJGQTKGU HCNN KPVQ VYQ ECVGIQTKGU CEEQTFKPI CU VQ YJGVJGT VJG[ VT[ VQ DNQEM VJG RCTCFQZ CV VJG HKTUV QT CV VJG UGEQPF UVGR 2TGFKECVKXKV[ TGUVTKEVKQPU DNQEM VJG HKTUV UVGR QP VJG TQCF VQ RCTCFQZ TWNKPI QWV VJG 4WUUGNN EQPEGRV +H QPG UVQRU CV VJG UKORNG RTGFKECVKXG NGXGN KV NQQMU CU KH QPG JCU CNNQYGF VJG ∉ KPVQ VJG NCPIWCIG DWV PQV CNNQYGF KV VQ FGVGTOKPG C EQPEGRV (QT VJCV HQTOWNC KU LWUV CP CDDTGXKCVKQP QH `∃ U YJKEJ KU CNNQYGF KP VJG UKORNG RTGFKECVKXG NCPIWCIG VJQWIJ EQORTGJGPUKQP KU PQV CUUWOGF HQT KV 0QY VJGTG KU PQ QDXKQWU TGCUQP YJ[ C RTGFKECVG QT QRGP HQTOWNC OC[ KP UQOG ECUGU NKMG C RTQRGT PCOG QT UKPIWNCT FGHKPKVG FGUETKRVKQP HCKN VQ JCXG C DWV KV KU XGT[ JCTF VQ UGG JQY QPG EQWNF TGCUQPCDN[ CNNQY HQTOWNCU KPVQ QPGOU NCPIWCIG VJCV QPG FQGU PQV CNNQY VQ JCXG +H QPG VJKPMU NKMG (TGIG QH VJG XCTKCDNGU CU TCPIKPI QXGT TGHGTGPVU KV UGGOU VYQ
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4WUUGNN HQTOWNC Z
Z
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:
TGHGTGPV
UGPUGU
:
:Z
Z
Z
Z
Z
Z
Z
VJGTG KU PQ QDXKQWU TGCUQP YJ[ VJGTG DG CP HQT GXGT[ φ DWV KH QPG VJKPMU NKMG 4WUUGNN QH VJG XCTKCDNGU TCPIKPI QXGT UGPUGU VJGTG FQGU UGGO VQ DG C TGCUQP VQ FGOCPF CP HQT GXGT[ φ 6JKU VJG TCOKHKGF RTGFKECVKXG VJGQT[ UWRRNKGU +P VJCV VJGQT[ VJGTG KU PQ NQPIGT C UKPING HQTOWNC `∃ U DWV C UGSWGPEG QH HQTOWNCU C XGTUKQP YKVJ UWRGTUETKRV VJCV FGVGTOKPGU CP YKVJ UWRGTUETKRV C XGTUKQP YKVJ UWRGTUETKRV VJCV FGVGTOKPGU CP YKVJ UWRGTUETKRV CPF UQ QP 6JGTG KU PQ HQTOWNC VJCV FQGU PQV FGVGTOKPG C EQPEGRV DWV VJGTG KU PQ 4WUUGNN HQTOWNC +P GKVJGT ECUG YJGVJGT QPG UVQRU CV VJG UKORNG NGXGN QT IQGU QP VQ TCOKH[ VJGTG KU PQ 4WUUGNN EQPEGRV n ∉ #P CNVGTPCVKXG YQWNF DG VQ CNNQY HWNN KORTGFKECVKXG EQORTGJGPUKQP DWV DNQEM VJG UVGR HTQO VJG 4WUUGNN EQPEGRV VQ VJG 4WUUGNN UGV D[ TGUVTKEVKPI VJG CUUWORVKQP VJCV VJGTG GZKUVU CP GZVGPUKQP HQT GXGT[ EQPEGRV 1PG QDXKQWU YC[ VQ KORNGOGPV VJKU KFGC HQTOCNN[ YQWNF DG VQ IQ DCEM VJG MKPF QH RTGUGPVCVKQP QH (TGIGOU UGV VJGQT[ VJCV JCU CP GZRNKEKV RTGFKECVG ] HQT QKU VJG GZVGPUKQP QHR 9G OC[ TGVCKP VJG CZKQO QH UGVGZVGPUKQPCNKV[ EJCTCEVGTK\KPI YJCV GZVGPUKQPU CTG NKMG YJGP VJG[ GZKUV PCOGN[ OWUV
:
Z
:
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:
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:
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Z:
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→ Z [ ↔ : ≡ ;
$[ KVUGNH QPN[ KORNKGU VJCV GZVGPUKQPU CTG WPKSWG YJGP VJG[ GZKUV (TGIG IQV KPVQ VTQWDNG D[ CUUWOKPI VJG CZKQO QH UGVEQORTGJGPUKQP ∃ ] CUUGTVKPI VJCV GZVGPUKQPU CNYC[U FQ GZKUV 6JG QDXKQWU YC[ VQ TGUVTKEV VJKU WPCEEGRVCDNG CZKQO QH YQWNF DG VQ TGRNCEG VJKU WPEQPFKVKQPCN GZKUVGPEG CZKQO D[ C EQPFKVKQPCN QT DKEQPFKVKQPCN XGTUKQP QH QPG QH VJG HQNNQYKPI VYQ HQTOU Z
C
σ(:) → ∃Z
]
D
Z:
∃Z
]
Z:
Z:
↔ σ(:)
*GTG σ YQWNF GZRTGUU C RTQXKUQ PQV OGV D[ VJG 4WUUGNN EQPEGRV VCMGP VQ DG UWHHKEKGPV
KP VJG EQPFKVKQPCN ECUG QT PGEGUUCT[ CPF UWHHKEKGPV KP VJG DKEQPFKVKQPCN ECUG VQ OCMG KV UCHG VQ CUUWOG VJG EQPEGRV JCU CP GZVGPUKQP 6JGTG KU CNUQ C NGUU QDXKQWU YC[ VQ VJKPM CDQWV TGUVTKEVKPI VJG CUUWORVKQP QH VJG GZKUVGPEG QH GZVGPUKQPU (QT YG JCXG UGGP VJCV VJG CUUWORVKQP QH VJG GZKUVGPEG QH GZVGPUKQPU KU GSWKXCNGPV VQ VJG CUUWORVKQP QH VJG GZKUVGPEG QH CDUVTCEVU HQT CNN GSWKXCNGPEGU VJG ECUG QH EQGZVGPUKXGPGUU ≡ UWDUWOKPI VJG ECUG QH CP[ QVJGT GSWKXCNGPEG #P CNVGTPCVKXG VQ CUUWOKPI UCHG GZVGPUKQPU YQWNF DG VQ CUUWOG PQ GZVGPUKQPU CV CNN DWV QPN[ CDUVTCEVU HQT UQOG QVJGT GSWKXCNGPEG QT GSWKXCNGPEGU #EVWCNN[ VJKU CRRTQCEJ OQTG QT NGUU UWDUWOGU VJG UCHGGZVGPUKQPU CRRTQCEJ CV NGCUV KP :
:
'
CDUVTCEVKQPKUV
VJG DKEQPFKVKQPCN XGTUKQP D (QT EQTTGURQPFKPI VQ CP[ UCHGV[ EQPFKVKQP σ YG JCXG CP GSWKXCNGPEG ≡σ QH FGHKPGF CU HQNNQYU :
UCHG EQGZVGPUKXGPGUU
:
≡σ ; ↔ : ≡ ; ∨ `σ : ` σ ;
6JKU ≡σ CITGGU YKVJ EQGZVGPUKXGPGUU HQT UCHG EQPEGRVU DWV NWORU CNN WPUCHG QPGU VQIGVJGT #UUWOKPI CDUVTCEVU HQT ≡σ KU GSWKXCNGPV VQ CUUWOKPI D RNWU VJG GZKUVGPEG QH QPG CFFKVKQPCN PCOGCDNG QDLGEV ο FKUVKPEV HTQO CNN GZVGPUKQPU VJG CDUVTCEV QH YKVJ TGURGEV VQ ≡σ COQWPVU VQ VJG GZVGPUKQP QH KH KU UCHG CPF VQ ο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
:
:
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3
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SWCPVKHKGTU YTKVKPI
∀[ [ ∈ Z → W
∃[ [ ∈ Z W
#PCNQIQWU VQ VJG CTKVJOGVKECN CPF CPCN[VKECN JKGTCTEJKGU VJGTG KU VJG
.GX[ JKGTCTEJ[
HQTOWNCU QH UGV VJGQT[ 6JQUG YKVJ QPN[ DQWPFGF SWCPVKHKGTU CTG ECNNGF
DQWPFGF
QT
QH
∆ QT
Σ QT Π 2TGHKZKPI GZKUVGPVKCN TGURGEVKXGN[ WPKXGTUCN SWCPVKHKGTU IKXGU Σ TGURGEVKXGN[ Π HQTOWNCU CPF VJG JKGTCTEJ[ EQPVKPWGU YKVJ ΣP CPF ΠP HQT P KP VJG QDXKQWU YC[ # UWDUETKRV
P
QP < YKNN KPFKECVG TGUVTKEVKPI UGRCTCVKQP VQ
ΣP HQTOWNCU φ CPF C UWDUETKRV P
QP <( YKNN KPFKECVG UKOKNCTN[ TGUVTKEVKPI TGRNCEGOGPV CU YGNN 9G YKNN UGG KP VJG PGZV EJCRVGT VJCV <
i CPF KPFGGF GZVGPUKQPCNKV[ PWNN UGV
RCKTKPI CFLWPEVKQP CPF UGRCTCVKQP KU UWHHKEKGPV VQ IKXG C U[UVGO KP YJKEJ HKTUVQTFGT 2GCPQ CTKVJOGVKE 2 ECP DG KPVGTRTGVGF +PXGTUGN[ < VJCV OCVVGT EJQKEG CU YGNN ECP DG KPVGTRTGVGF KP 2
WUKPI C EQFKPI QH HKPKVG UGVU D[ PCVWTCN
PWODGTU 1VJGT RCKTU QH VJGQTKGU CTG UKOKNCTN[ TGNCVGF < (
℘ ω CPF 2 < ( ℘P ω CPF 2P
PQV PGGF TGRNCEGOGPV VQ KPVGTRTGV KPVGTRTGV UGV VJGQT[ KP 2P
<
2P
i CPF KPFGGF TGRNCEGOGPV CPF HQT CPF
ΠP %# < ( CPF 2
6JG RCTGPVJGVKECN ( KPFKECVGU VJCV YG FQ
KP UGV VJGQT[ DWV FQ IGV TGRNCEGOGPV YJGP YG
KU UNKIJVN[ UVTQPIGT VJCP VJG UKORNG VJGQT[ QH V[RGU 2ω <( KU
EQPUKFGTCDN[ UVTQPIGT VJCP < VJQWIJ QPG JCU VQ NQQM NQPI CPF JCTF VQ HKPF CP[ VJGQTGO QH OCVJGOCVKECN CPCN[UKU QH KPVGTGUV VQ YQTMKPI OCVJGOCVKEKCPU YJGTG VJG FKHHGTGPEG DGVYGGP VJG VYQ U[UVGOU YQWNF OCMG C FKHHGTGPEG /CP[ GZVGPUKQPU QH <( D[ VJG CFFKVKQP QH HWTVJGT CZKQOU VWTP QWV VQ DG KPVGTRTGVCDNG KP <( KVUGNH 0QVCDN[ D[ TGUWNVU QH 2CWN %QJGP CUUWOKPI <( KU EQPUKUVGPV #% KU PQV C VJGQTGO QH KV 0QT KU VJG
EQPVKPWWO J[RQVJGUKU
%* CUUGTVKPI VJCV VJG PWODGT
c QH TGCN PWODGTU KU VJG PGZV NCTIGT ECTFKPCN ℵ CHVGT VJG PWODGT ℵ QH PCVWTCN PWODGTU C VJGQTGO QH <(% <( #% 0QT KU KVU IGPGTCNK\CVKQP )%* C VJGQTGO QH <(% %* 0QT
O
KU )ÑFGN U
CZKQO QH EQPUVTWEVKDKNKV[
8 . YJQUG HQTOWNCVKQP PGGF PQV FGVCKP WU C
VJGQTGO QH <(% )%* ;GV D[ GCTNKGT TGUWNVU QH )ÑFGN <(% 8 . KU KPVGTRTGVCDNG KP <( CPF OQTGQXGT KORNKGU )%* JGPEG %* CU YGNN CU #% 6JGUG TGUWNVU CNUQ CRRN[ OWVCVKU OWVCPFKU
VQ VJG UWDVJGQTKGU QH < CPF <( EQPUKFGTGF KP VJG RTGEGFKPI RCTCITCRJ
G
.CTIG %CTFKPCNU
6JG UKORNG RTGFKECVKXG UGEQPFQTFGT XGTUKQP QH <(% KU MPQYP CU 0$) QT 0GWOCPP $GTPC[U )ÑFGN UGV VJGQT[
O
+V KU C EQPUGTXCVKXG GZVGPUKQP D[ 5JQGPHKGNF U
VJGQTGO 6JG UGEQPFQTFGT VJGQT[ <(% KU IGPGTCNN[ MPQYP CU /- QT VJGQT[
XQP
/QTUG -GNN[ UGV
+P DQVJ VJG RTGFKECVKXG CPF HWNN UGEQPFQTFGT XGTUKQPU VJG UEJGOG QH UGRCTCVKQP KU
TGRNCEGF D[ C UKPING CZKQO VJWU
∃Z∀[ [ ∈ Z ↔ [ ∈ W 7[
5GRCTCVKQP
#PF UKOKNCTN[ HQT TGRNCEGOGPV /- KU UVTQPIGT VJCP <(% DWV QPN[ UNKIJVN[ UQ /QTG UKIPKHKECPVN[ UVTQPIGT VJGQTKGU CTKUG YJGP QPG CFFU VQ <(% EGTVCKP QT
KPHKPKV[
NCTIG ECTFKPCN
JKIJGT
CZKQOU 6JGUG CUUGTV VJG GZKUVGPEG QH UQOG XGT[ NCTIG UGVU
VJGTGD[ KORN[KPI VJG GZKUVGPEG QH UQOG XGT[ NCTIG ECTFKPCN PWODGTU 6JG HQTOWNCVKQP QH VJGUG ECPFKFCVG CZKQOU KU KP CNN ECUGU TCVJGT VGEJPKECN VJQWIJ KV KU NGCUV UQ KP VJG ECUG QH UQOG QH VJG YGCMGT QPGU COQPI VJGO # UGV
/
KU ECNNGF
ECPFKFCVG OQFGN
TGNCVKQP UQ VJCV
VTCPUKVKXG
KH
[
∈ / YJGPGXGT Z ∈ / CPF [ ∈ Z # PCVWTCN
HQT <(% JCU C VTCPUKVKXG UGV
/
CU KVU FQOCKP CPF VJG CEVWCN GNGOGPVJQQF
∈ COQPI GNGOGPVU QH / CU VJG FGPQVCVKQP CUUKIPGF VQ VJG GNGOGPVJQQF U[ODQN ∈
C D
∈ / UCVKUH[ Z ∈ [ KH CPF QPN[ KH C ∈ D 6TCPUKVKXKV[ IWCTCPVGGU VJCV
GZVGPUKQPCNKV[ YKNN DG VTWG KP UWEJ C ECPFKFCVG OQFGN KH GNGOGPVU FKUVKPEV VJGTG KU UQOG
E
C
CPF
D
QH
/
CTG
VJCV KU CP GNGOGPV QH QPG DWV PQV VJG QVJGT QH VJGO CPF VJKU
D[ VTCPUKVKXKV[ KVUGNH DG CP GNGOGPV QH
/
CPF
C
CPF
D
CPF
E
YKNN UCVKUH[ ` \
E
YKNN
∈ Z ↔ \ ∈ [
KP VJG OQFGN )GPGTCNN[ URGCMKPI YG IGV C OQFGN QH VJG QVJGT CZKQOU QPN[ KH
/
HWNHKNNU
UQOG CFFKVKQPCN ENQUWTG EQPFKVKQPU EQPVCKPU VJG PWNN UGV EQPVCKPU VJG WPQTFGTGF RCKT QH CP[ UGVU KV EQPVCKPU CPF UQ QP # UGV CPF
[
/
KU ECNNGF
⊆ Z # PCVWTCN
UWRGTVTCPUKVKXG
KH
/
KU VTCPUKVKXG CPF KH
ECPFKFCVG OQFGN HQT <(%
[
∈ / YJGPGXGT Z ∈ /
/- JCU C UWRGTVTCPUKVKXG UGV
FQOCKP QH KVU HKTUVQTFGT XCTKCDNGU CPF VJG RQYGT UGV
/
℘ / CU VJG FQOCKP QH KVU
CU VJG
UGEQPFQTFGT XCTKCDNGU YKVJ #
C D
∈ / UCVKUH[KPI Z ∈ [ KH CPF QPN[ KH C ∈ D CPF C ∈ / CPF
⊆ / UCVKUH[KPI :Z KH CPF QPN[ KH C ∈ # 5WRGTVTCPUKVKXKV[ IWCTCPVGGU VJCV VJG UKPING
CZKQO XGTUKQP QH UGRCTCVKQP CDQXG YKNN DG VTWG KP UWEJ C ECPFKFCVG OQFGN <(% KH EQPUKUVGPV ECPPQV RTQXG VJCV CP[ PCVWTCN ECPFKFCVG OQFGN KU KPFGGF C OQFGN QH <(% HQT VJCV YQWNF RTQXG VJCV <(% JCU C OQFGN CPF VJGTGHQTG YQWNF RTQXG VJCV <(% EQPUKUVGPV CPF PQ EQPUKUVGPV VJGQT[ ECP RTQXG KVU QYP EQPUKUVGPE[ 5KOKNCTN[ YJKNG <(% ECP RTQXG VJCV VJGTG KU C PCVWTCN OQFGN QH <(% KV ECPPQV RTQXG VJCV CP[ PCVWTCN ECPFKFCVG OQFGN QH <(% KU KPFGGF C OQFGN QH <(% 6JG J[RQVJGUKU QH VJG GZKUVGPEG QH CP KPCEEGUUKDNG ECTFKPCN
UKORN[ DG VCMGP VQ
DG
KU GSWKXCNGPV VQ CPF HQT RTGUGPV RWTRQUGU OC[
VJG J[RQVJGUKU VJCV VJGTG GZKUVU C PCVWTCN OQFGN QH <(% 6JG
ECTFKPCN PWODGT QH UWEJ C UGV KU ECNNGF CP
KPCEEGUUKDNG
GZKUVGPEG QH EQHKPCNN[ OCP[ KPCEEGUUKDNG ECTFKPCNU
ECTFKPCN 6JG J[RQVJGUKU QH VJG
OC[ DG VCMGP VQ DG VJG J[RQVJGUKU VJCV
GXGT[ UGV KU CP GNGOGPV QH UQOG PCVWTCN OQFGN QH <(% 6JG J[RQVJGUKU QH VJG C J[RGTKPCEEGUUKDNG ECTFKPCN
GZKUVGPEG QH
OC[ DG VCMGP VQ DG VJG J[RQVJGUKU VJCV VJGTG GZKUVU C PCVWTCN
OQFGN QH <(% EQHKPCNN[ OCP[ KPCEEGUUKDNGU 6JG ECTFKPCN PWODGT QH UWEJ C UGV KU ECNNGF C
J[RGTKPCEEGUUKDNG
NCPIWCIG QH <(% CDQWV
OQFGNU
ECTFKPCN #NN VJGUG J[RQVJGUGU ECP DG HQTOWNCVGF KP VJG
M VJG UGEQPFQTFGT XCTKCDNGU QH <(% CTG PQV PGGFGF UKORN[ VQ VCNM
QH UGEQPFQTFGT VJGQTKGU
+PCEEGUUKDNGU J[RGTKPCEEGUUKDNGU UWRGTJ[RGTKPCEEGUUKDNGU CPF UQ QP CTG QPN[ VJG
M UWEJ CU VJG ECTFKPCNU QH YJKEJ VJGTG KU C JKGTCTEJ[ QH MKPFU M GXGP COQPI
HKTUV CPF UOCNNGUV QH NCTIG ECTFKPCNU 6JGTG CTG OCP[ NCTIGT NCTIG ECTFKPCNU UQECNNGF
KPFGUETKDCDNG
VJG UQECNNGF
UOCNN
NCTIG ECTFKPCNU VJQUG YJQUG GZKUVGPEG KU EQPUKUVGPV YKVJ QT RGTJCRU +
UJQWNF UC[ PQV MPQYP VQ DG KPEQPUKUVGPV YKVJ 8 . 1PG UGEQPFQTFGT UGV VJGQT[ YJQUG UVTGPIVJ QEEWRKGU C RQUKVKQP KP VJG JKGTCTEJ[ QH KPFGUETKDCDNGU
$GTPC[U UGV VJGQT[
$ YKNN
RNC[ C URGEKCN TQNG VQYCTFU VJG GPF QH VJKU OQPQITCRJ DWV + YKNN FGHGT FGUETKRVKQP QH KV WPVKN VJGP $G[QPF VJGUG CTG VJG NCTIG NCTIG ECTFKPCNU VJGOUGNXGU FKXKFGF KPVQ VJG OGFKWO NCTIG NCTIG ECTFKPCNU EQPUKUVGPV YKVJ C OQFKHKGF XGTUKQP QH 8 . CPF VJG GZVTC NCTIG NCTIG ECTFKPCNU 6JG UVTQPIGUV NCTIG ECTFKPCN J[RQVJGUKU QH CNN VJG GZKUVGPEG QH C 4GKPJCTFV ECTFKPCN
JCU DGGP UJQYP VQ DG KPEQPUKUVGPV CNN VJG TGUV CTG RTGUWOGF EQPUKUVGPV
D[ UGV VJGQTKUVU VJQWIJ YKVJ FGETGCUKPI EQPHKFGPEG CU QPG IQGU WR VJG UECNG 'CEJ PGY J[RQVJGUKU KORNKGU PGY VJGQTGOU GZRTGUUKDNG KP VJG NCPIWCIG QH HKTUVQTFGT CTKVJOGVKE CPF KPFGGF VJGQTGOU GZRTGUUKDNG D[
Π
HQTOWNCU
6JG UECNG HTQO 4QDKPUQP VQ 4GKPJCTFV KU UWOOCTK\GF KP 6#$.' ' YJGTGKP KPFKECVGU OWVWCN KPVGTRTGVCDKNKV[ QT GSWCN EQPUKUVGPE[ UVTGPIVJ 9G YKNN GPEQWPVGT PGQ(TGIGCP U[UVGOU CV VJG NGXGNU QH 3 QT 3 +∆ QH 3 +∆ GZR QH (
≈
Π %#
QH 2 2# QH 2 2# CPF QH $ 'XKFGPVN[ UQOG PGQ(TGIGCP U[UVGOU CTG OQTG NKOKVGF VJCP QVJGTU CU VQ JQY OWEJ QH ENCUUKECN OCVJGOCVKEU ECP DG FGXGNQRGF YKVJKP VJGO
2JKNQUQRJKECN6CTIGVUHQT0GQ(TGIGCPKUO
#ICKPUV CP[ NKOKVCVKQPU QP JQY OWEJ OCVJGOCVKEU QPG ECP IGV QP C IKXGP PGQ(TGIGCP CRRTQCEJ OWUV DG YGKIJGF CP[ RJKNQUQRJKECN DGPGHKVU CVVCEJKPI VQ QDVCKPKPI UC[ VJG PCVWTCN PWODGTU KP VJG YC[ VJCV RCTVKEWNCT PGQ(TGIGCP CRRTQCEJ FQGU 9JCV DGPGHKVU KH CP[ VJGTG CTG YKNN FGRGPF QP VJG FGVCKNU QH VJG RCTVKEWNCT PGQ(TGIGCP UVTCVGI[ KP SWGUVKQP DWV UQOG IGPGTCN TGOCTMU OC[ DG KP QTFGT KP CFXCPEG DGHQTG WPFGTVCMKPI VJG RTQOKUGF UWTXG[ QH UVTCVGIKGU 6Q DGIKP YKVJ QPG OWUV FKUVKPIWKUJ VYQ
MKPFU
QH
CFXCPVCIGU VJCV OKIJV DG ENCKOGF HQT C IKXGP PGQ(TGIGCP CRRTQCEJ QT VYQ RTQDNGOU VQ YJQUG UQNWVKQP VJCV CRRTQCEJ OKIJV DG ENCKOGF VQ EQPVTKDWVG 6JG HKTUV KU VJG KPHQTOCVKQP
KFGPVKV[
RTQDNGO 9JCV CTG PCVWTCN PWODGTU! 6JG UGEQPF KU VJG
RTQDNGO *QY ECP YG JCXG MPQYNGFIG QH PCVWTCN PWODGTU! (TGIG CNTGCF[
JKOUGNH JCF DGGP EQPEGTPGF KP JKU QYP YC[ YKVJ DQVJ RTQDNGOU *KU HCOQWU SWGUVKQP
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
GZGIGVKECN
ENCKO VJCV VJG FGHKPKVKQP EQTTGEVN[ UVCVGU YJCV
QPCVWTCN PWODGTR
TGCNN[ OGCPV CNN CNQPI QT KP VJG OCVGTKCN TCVJGT VJCP VJG HQTOCN OQFG YJCV PCVWTCN PWODGTU TGCNN[ YGTG CNN CNQPI UQ VJCV VJG KFGPVKV[ RTQDNGO KU UQNXGF 6JG QVJGT YQWNF DG VJG
GRKUVGOQNQIKECN
ENCKO VJCV KH YG VCMG PCVWTCN PWODGTU VQ DG KPVTQFWEGF D[ VJG
PGQ(TGIGCP FGHKPKVKQP VJGP KV DGEQOGU GCUKGT VQ GZRNCKP JQY YG ECP JCXG MPQYNGFIG QH VJGO CPF UQ UQNXG VJG KPHQTOCVKQP RTQDNGO 6JG GZGIGVKECN CPF GRKUVGOQNQIKECN KUUWGU CTG NCTIGN[ KPFGRGPFGPV QH GCEJ QVJGT *QY VJGTG OKIJV DG CP CFXCPEG QP VJG GRKUVGOQNQIKECN HTQPV YKVJQWV CP[ RTQITGUU QP GZGIGVKECN KUUWGU KU KNNWUVTCVGF D[ VJG JKUVQT[ QH FGHKPKVKQPU QH KOCIKPCT[ QT EQORNGZ PWODGTU )TQUUN[ QXGTUKORNKH[KPI HQT VJG UCMG QH GZCORNG YG OC[ TGFWEG VJCV JKUVQT[ VQ
LWUV VYQ UVCIGU KPXQNXKPI LWUV VYQ HKIWTGU VJTGG EGPVWTKGU CRCTV ECTVQQP XGTUKQPU QH %CTFCPQ CPF *COKNVQP 1WT ECTVQQP %CTFCPQ DGIKPU YKVJ VJG CNIGDTC QH TGCN PWODGTU JCXKPI U[ODQNU HQT VJG CFFKVKQP CPF OWNVKRNKECVKQP CPF OCP[ NCYU CDQWV VJGO *G CUUWOGU VJCV CFFKVKQP CPF OWNVKRNKECVKQP CPF VJGKT KPXGTUGU UWDVTCEVKQP CPF FKXKUKQP ECP UVKNN DG CRRNKGF CPF YKNN UVKNN QDG[ VJG UCOG NCYU KH C PGY QDLGEV K VQ DG ECNNGF VJG CPF CUUWOGF VQ HWNHKNNGF VJG EQPFKVKQP QDLGEVU VQ DG ECNNGF VJG
KOCIKPCT[ WPKV
KU CFLQKPGF
⋅ K VJWU QDVCKPKPI C YJQNG U[UVGO QH PGY
K
EQORNGZ PWODGTU
6JG EQORNGZ PWODGTU UQQP RTQXG VQ DG
OCVJGOCVKECNN[ WUGHWN KP VJCV CTIWOGPVU KPXQNXKPI KOCIKPCT[ PWODGTU [KGNF EQPENWUKQPU CDQWV TGCN PWODGTU VJCV ECP DG XGTKHKGF VJQWIJ UKPEG PQ LWUVKHKECVKQP HQT VJG QTKIKPCN CUUWORVKQPU KU QHHGTGF QPG JCU QPN[ KPFWEVKXG GXKFGPEG VJCV VJG EQPENWUKQPU CDQWV TGCN PWODGTU VJWU QDVCKPGF YKNN CNYC[U DG EQTTGEV 1WT ECTVQQP *COKNVQP DGIKPU YJGTG %CTFCPQ NGCXGU QHH DWV CVVCEJGU C URGEKCN KORQTVCPEG VQ VJTGG QH VJG OCP[ NCYU QH EQORNGZ PWODGTU
∀\ \ EQORNGZ → ∃Z∃[ Z TGCN [ TGCN \ Z [K
Z
[K
W
Z
[K
⋅ W XK ZW [X ZX [W K
*COKNVQP RTQRQUGU VQ TGCN PWODGTU CPF VQ
XK
Z
FGHKPG
FGHKPG
W
[
X K
VJG EQORNGZ PWODGT
Z
[K
VQ DG UKORN[ QTFGTGF RCKT
〈Z [〉 QH
CFFKVKQP CPF OWNVKRNKECVKQP HQT EQORNGZ PWODGTU KP VGTOU QH
CFFKVKQP CPF OWNVKRNKECVKQP QH TGCN PWODGTU VJWU
↔ ∃Z∃[ Z TGCN [ TGCN \ 〈Z [〉 〈Z [〉 〈W X〉 〈Z [ W X〉
〈Z [〉 ⋅ 〈W X〉 〈ZW
\
EQORNGZ
[X ZX
〉
[W
6JGP CPF HQNNQY CPF OQTGQXGT JG KU CDNG VQ UJQY VJCV HTQO VJGO HQNNQY IKXGP VJG NCYU QH TGCN PWODGTU CPF QTFGTGF RCKTU CNN VJG NCYU QH EQORNGZ PWODGTU QH CP[ KORQTVCPEG KP OCVJGOCVKEU RWTG QT CRRNKGF +V KU ENGCT VJCV *COKNVQP JCU OCFG CP
GRKUVGOQNQIKECN
CFXCPEG QXGT %CTFCPQ
FKURQUKPI QH CP[ KPHQTOCVKQP RTQDNGO CDQWV EQORNGZ PWODGTU QXGT CPF CDQXG VJG
O
KPHQTOCVKQP RTQDNGO HQT TGCN PWODGTU CPF VJCV HQT QTFGTGF RCKTU $WV CTG *COKNVQP U FGHKPKVKQPU CEEGRVCDNG CU
GZGIGUKU
QH %CTFCPQ! # TGCUQP HQT VJKPMKPI PQV DGEQOGU
CRRCTGPV YJGP QPG EQPUKFGTU QVJGT YC[U KP YJKEJ *COKNVQP EQWNF JCXG RTQEGGFGF *G EQWNF JCXG KFGPVKHKGF VJG EQORNGZ PWODGT
Z
[K
[K
Z
YKVJ VJG QTFGTGF RCKT
〈[ Z〉
7UKPI C FKHHGTGPV PQTOCN HQTO VJGQTGO JG EQWNF JCXG KFGPVKHKGF C EQORNGZ PWODGT T EQU
θ K UKP θ YKVJ VJG QTFGTGF RCKT 〈T θ〉 UWDLGEV VQ CRRTQRTKCVG TGUVTKEVKQPU QP T CPF
θ QT HQT VJCV OCVVGT YKVJ 〈θ T〉 9KVJ CRRTQRTKCVG OQFKHKECVKQPU QH CPF QPG CICKP IGVU CPF CPF VJGPEG CNN QVJGT NCYU QH CP[ KORQTVCPEG 6JG KFGPVKHKECVKQP YKVJ
〈T θ〉 QT 〈θ T〉 KU UQ CTVKHKEKCN VJCV YG OC[ DG UWTG VJCV
O
%CTFCPQ FKF PQV JCXG KV KORNKEKVN[ KP OKPF *COKNVQP U KFGPVKHKECVKQP YKVJ
〈Z [〉 ECPPQV DG
KOOGFKCVGN[ FKUOKUUGF QP ITQWPFU QH CTVKHKEKCNKV[ DWV VJGP PGKVJGT ECP VJG CNVGTPCVKXG KFGPVKHKECVKQP YKVJ
〈[ Z〉 5KPEG DQVJ KFGPVKHKECVKQPU [KGNF CNN NCYU QH CP[ KORQTVCPEG VQ C
OCVJGOCVKEKCP NKMG %CTFCPQ GXGP DGHQTG TGCFKPI JKU YQTMU YG ECP RTGFKEV VJCV YG YKNN HKPF YJCV YG FQ HKPF YJGP YG NQQM CV VJGO PCOGN[ VJCV VJGTG KU C EQORNGVG NCEM QH GXKFGPEG KP VJGO VQ UJQY YJKEJ QH VJG VYQ QTFGTGF RCKTU JG WPFGTUVQQF
Z
[K
[K
VQ
Z
DG ETGCVKPI C UVTQPI FQWDV YJGVJGT JG OGCPV EQORNGZ PWODGTU VQ DG CP[ UQTV QH QTFGTGF RCKTU CV CNN /QTGQXGT VJGTG KU C UKOKNCT NCEM QH GXKFGPEG VJCV JG TGICTFGF JCU JCXKPI CP[ URGEKCN
QFGHKPKVKQPCNR UVCVWU COQPI VJG \KNNKQP NCYU QH EQORNGZ PWODGTU VJCV
HQNNQY HTQO JKU KPKVKCN CUUWORVKQPU
QPCVWTCNR HQT QEQORNGZR UWDUVKVWVG (TGIGOU CPF 4WUUGNNOU CPF
Z
CPF
[
〉
[ Z
〈T θ〉 CPF 〈θ T〉 UWDUVKVWVG VJG 2GCPQ RQUVWNCVGU HQT CPF CPF [QW JCXG VJG EQTG
O
QH VJG CPVKGZGIGVKECN CTIWOGPV KP $GPCEGTTCH +H $GPCEGTTCH U CTIWOGPVU CTG CEEGRVGF QPN[ VYQ QRVKQPU TGOCKP VQ VCMG PCVWTCN PWODGTU CU QDLGEVU CFQRV VJG
UVTWEVWTCNKUV
CUUGTVKQPU CDQWV
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m
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∀Z ν Z → φν Z W
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∀ ν W → φ W ∀ ν → φ W → φ W → ∀ ν → φ W W
Z
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QT TGXGTVKPI VQ VJG RTCEVKEG QH UWRRTGUUKPI RCTCOGVGTU VJG HQNNQYKPI
φ ∀Z ν Z → φ Z → φ Z → ∀Z ν Z → φ Z
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Y
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∃[ ²[ ∀Z Z ∈ [ ↔ φ Z ↔ #Z φ Z
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∀[(²[ → ∃\ ²\ ∀Z Z ∈ \ ↔ Z ∈ [ ψ Z
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m
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∃W∀[ [ ∈ W ↔ [
m
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m
m
m
m
m
m
HQT GXGT[ PCVWTCN PWODGT Z NGV Z DG CP[ PCVWTCN PWODGT CPF CRRN[ UGRCTCVKQP VQ =Z? VQ QDVCKP C UGV W UWEJ VJCV HQT CNN Y YG JCXG
Y
∈W↔Y
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Z
φ Y
m
+ ENCKO W KU KPFWEVKXG WR VQ Z 9G JCXG ∈ W UKPEG Z KU C PCVWTCN PWODGT CPF VJGTGHQTG Z CPF UKPEG φ D[ QWT J[RQVJGUKU CDQWV φ 5WRRQUG YG JCXG Y ∈ W CPF Y Z 6Q RTQXG Y ∈ W HKTUV PQVG VJCV UKPEG Y ∈ W YG JCXG φ Y 6JGP PQVG VJCV UKPEG Y Y Z CPF Z KU C PCVWTCN PWODGT UQ CTG Y CPF Y 6JGTGHQTG φ Y JQNFU D[ QWT J[RQVJGUKU CDQWV φ CPF JGPEG Y ∈ W CU TGSWKTGF VQ UJQY W KPFWEVKXG WR VQ Z 5KPEG Z KU KPFWEVKDNG YG JCXG Z ∈ W CPF VJGTGHQTG φ Z 6JWU VJG HWNN WPTGUVTKEVGF RTKPEKRNG QH OCVJGOCVKECN KPFWEVKQP HQT CNN HQTOWNCU QH VJG NCPIWCIG CRRNKGU VQ PCVWTCN PWODGTU 9JCV YGTG KP VJG RTQQH QH VJG KPVGTRTGVCDKNKV[ QH 3 OGTGN[
m m
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O
W Q
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R
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∃:∀Z :Z ↔ φ Z YJGTG
φ Z EQPVCKPU PQ DQWPF EQPEGRV XCTKCDNGU
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: ; ↔ : ≈ ;
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DG VJG HWNN RQYGT UGV QT UGV QH CNN UWDUGVU QH VJG FQOCKP QH VJG QDLGEV OQFGN CPF KU C OQFGN QH HWNN KORTGFKECVKXG EQORTGJGPUKQP
6Q TGVWTP PQY VQ VJG EQPUGTXCVKXGPGUU ENCKO UVCVGF CDQXG NGV HKTUV ENCKO VJCV KH NQIKE VQ
6
6
KU EQPUKUVGPV VJGP VJG TGUWNV
KU CNUQ EQPUKUVGPV (QT KH
6
6
DG C HKTUVQTFGT VJGQT[ +
QH CFFKPI HWNN KORTGFKECVKXG UGEQPFQTFGT
KU EQPUKUVGPV D[ VJG )ÑFGN EQORNGVGPGUU VJGQTGO KV JCU C
OQFGN #U LWUV GZRNCKPGF VJKU OQFGN ECP DG VWTPGF KPVQ C OQFGN QH 6
6
6
M C UVCPFCTF OQFGN 5KPEG
VJWU JCU C OQFGN KV KU EQPUKUVGPV CU ENCKOGF 6JG EQPUGTXCVKXGPGUU TGUWNV HQNNQYU KOOGFKCVGN[
HTQO VJG EQPUKUVGPE[ TGUWNV (QT KH VJGP
6
φ KU C HQTOWNC QH VJG NCPIWCIG QH 6 VJCV KU PQV RTQXCDNG KP 6
`φ KU EQPUKUVGPV CPF CRRN[KPI VJG EQPUKUVGPE[ TGUWNV VQ KV KV HQNNQYU VJCV
EQPUKUVGPV CPF
6
`φ KU
φ KU PQV RTQXCDNG KP 6
6JG HQTGIQKPI CTIWOGPV OC[ HKTUV JCXG DGGP OCFG D[ *CQ 9CPI YJQ EQPUKFGTGF UGXGTCN GZCORNGU QH C VJGQT[
6
EQPUKUVKPI QH HWNN KORTGFKECVKXG OQPCFKE UGEQPFQTFGT NQIKE
QXGT C HKTUVQTFGT UGV VJGQT[
6
+PFGGF YG OC[ ECNN UWEJ VJGQTKGU CU
YKVJ CV NGCUV CU OWEJ LWUVKEG CU YG ECP ECNN VJGO
\KI\CI
6 9CPIUV[NG
QHNQCVKPIR
UGV VJGQTKGU
VJGQTKGU
6JG DGUVMPQYP 9CPIUV[NG UGV VJGQT[ KU KPFGGF QPG DCUGF QP C EQPUKFGTCDN[ UVTQPIGT WPFGTN[KPI HKTUVQTFGT UGV VJGQT[ VJCP CP[ QH VJG YGCM QPGU YG JCXG EQPUKFGTGF JGTG +V KU UGEQPFQTFGT UGV VJGQT[ VJCV 9CPI RTQRQUGF CU C TGRNCEGOGPV HQT VJG KPEQPUKUVGPV U[UVGO /. KPVTQFWEGF KP VJG HKTUV GFKVKQP QH 3WKPGOU VJGQT[ KU 3WKPGOU U[UVGO 0( QT
/CVJGOCVKECN .QIKE
Q0GY (QWPFCVKQPUR
+P VJKU ECUG VJG HKTUVQTFGT UGV
0( JCU KPVGTGUVGF C PWODGT QH YTKVGTU DGECWUG KV CNNQYU HQT C WPKXGTUCN UGV TGURGEV KV KU C \KI\CI TCVJGT VJCP C NKOKVCVKQP QH UK\G VJGQT[
M KP VJKU
M VJQWIJ YG JCXG UGGP C EQWRNG QH
QVJGT UKORNGT UGV VJGQTKGU 2( CPF 72( VJCV FQ UQ CU YGNN CPF YKNN UGG CPQVJGT KP VJG PGZV EJCRVGT 6JG KFGC WPFGTN[KPI VJG EJQKEG QH GZKUVGPEG CZKQOU KP 0( KU WP(TGIGCP /QTGQXGT HQT 0( VJG EQPUKUVGPE[ QH VJG U[UVGO TGOCKPU KP UGTKQWU FQWDV VQ VJKU FC[ +V JCU PQV DGGP RTQXGF EQPUKUVGPV TGNCVKXG VQ CP[ VJGQT[ KP YJQUG EQPUKUVGPE[ VJGTG KU CP[ XGT[ IGPGTCN EQPHKFGPEG COQPI URGEKCNKUVU (QT VJGUG TGCUQPU + YKNN NGCXG 0( CPF OQTG IGPGTCNN[ 9CPIUV[NG UGV VJGQTKGU CPF VJG \KI\CI KFGC CUKFG JGTG CPF TGVWTP VQ VJG RTQRGT UWDLGEV QH VJKU EJCRVGT
RTGFKECVKXG
VJGQTKGU
%QPUKUVGPE[ /QFGN6JGQTGVKE2TQQHU 1PG OKIJV YGNN JQRG VJCV KH QPG TQWPF QH RTGFKECVKXG EQPEGRVU IGVU WU 0GNUQP9KNMKG CTKVJOGVKE 3 YKVJ CFFKVKQP CPF OWNVKRNKECVKQP CPF KH VYQ TQWPFU IGVU WU -CNOCT CTKVJOGVKE 3
YKVJ GZRQPGPVKCVKQP VJGP VJTGG TQWPFU YQWNF IGV WU )GPV\GP CTKVJOGVKE 3 YKVJ UWRGTGZRQPGPVKCVKQP CPF VJCV KH YG EQPVKPWGF VJTQWIJ VJG YJQNG JKGTCTEJ[ QH C TCOKHKGF RTGFKECVKXG VJGQT[ YG YQWNF IGV CNN RTKOKVKXG TGEWTUKXG HWPEVKQPU 0QVJKPI QH VJG UQTV KU VJG ECUG HQT C TGCUQP RQKPVGF QWV VQ VJG CWVJQT KP C FKHHGTGPV EQPVGZV D[ 5CWN -TKRMG VJCV KU ENQUGN[ EQPPGEVGF YKVJ VJG TGCUQP YJ[ + JCXG CVVCEJGF )GPV\GPOU PCOG VQ 3 6JG TGCUQP KU UKORN[ VJKU VJCV VJG
EQPUKUVGPE[
QH VJG VJGQTKGU 28 CPF 28 CPF VJG
CPCNQIQWU 28 28
W CU YGNN CU VJCV QH VJG VJGQTKGU KP VJG RCTCNNGN UGTKGU DGIKPPKPI YKVJ 2*2 ECP DG RTQXGF WUKPI QHKPKVKUVKER RTQQHVJGQTGVKE OGVJQFU RKQPGGTGF D[ )GPV\GP YJKEJ CTG MPQYP VQ DG HQTOCNK\CDNG KP 3 $WV )ÑFGNOU UGEQPF EQORNGVGPGUU VJGQTGO UC[U TQWIJN[ URGCMKPI VJCV
PQ EQPUKUVGPV VJGQT[ KPVGTRTGVCDNG KP
ECP RTQXG KVU QYP EQPUKUVGPE[ CPF KORNKGU VJCV PQ VJGQT[ VJCV KU
6
ECP RTQXG VJG EQPUKUVGPE[ QH
6
ECP DG RTQXGF KP 3M VJGP VJG 3O HQT
O
6
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6
n M ECPPQV DG KPVGTRTGVGF KP 6 +P VJKU YC[ YG IGV WRRGT
DQWPFU QP JQY NCTIG C HTCIOGPV QH CTKVJOGVKE ECP DG KPVGTRTGVGF KP QWT RTGFKECVKXG (TGIGCP U[UVGOU
QKPHKPKUVKER OQFGNVJGQTGVKE EQPUKUVGPE[ RTQQHU HQT VJG
+P VJKU UGEVKQP + YKNN FGUETKDG
XCTKQWU RTGFKECVKXG (TGIGCP U[UVGOU KP VJG PGZV + YKNN FGUETKDG YJCV PGGFU VQ DG FQPG VQ EQPXGTV
QHKPKVKUVKER RTQQHVJGQTGVKE EQPUKUVGPE[ RTQQHU /[ IQCN KP VJKU UGEVKQP YKNN DG VQ
VJGUG KPVQ
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M C U[UVGO QHVGP ECNNGF (TGIG CTKVJOGVKE M HQT YJKEJ C UKORNG OQFGNVJGQTGVKE
EQPUKUVGPE[ RTQQH YKNN DG FKUEWUUGF KP VJG PGZV EJCRVGT
6JG EQPUKUVGPE[ QH 28 CPF 28 YKNN DG RTQXGF KP VJKU UGEVKQP D[ RTQXKPI VJG EQPUKUVGPE[ QH EGTVCKP CUUQEKCVGF KORTGFKECVKXG VJGQTKGU EQPXGPKGPV VQ VJKPM QH QDLGEV XCTKCDNGU
℘8 CPF ℘
8 YJKEJ + OWUV PQY KPVTQFWEG +V KU
℘8 CU DGKPI QDVCKPGF KP VYQ UVCIGU (KTUV NGV .
Z [ \
W CPF HKTUVTQWPF EQPEGRVXCTKCDNGU :
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;
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DG VJG NCPIWCIG YKVJ
W CPF LWUV VJG KFGPVKV[
DG VJG VJGQT[ YJQUG CZKQOU CTG LWUV VJG KPUVCPEGU QH VJG HWNN
6
KORTGFKECVKXG EQORTGJGPUKQP UEJGOG
∃: ∀Z : Z ↔ φ Z YJGTG φ Z KU C HQTOWNC QH .
φ Z KU RGTOKVVGF VQ EQPVCKP DQWPF HKTUVTQWPF EQPEGRV XCTKCDNGU 0QY NGV . CFF VJG GZVGPUKQP U[ODQN U CPF NGV 6 ℘8 CFF .CY 8
0QVG VJCV
U
:
U
;
↔ ∀\ :
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↔;
\
0QY IKXGU HWNN KORTGFKECVKXG EQORTGJGPUKQP HQT HQTOWNCU RQKPVU QWV VJCV HQT HQTOWNCU
YKVJQWV DQWPF EQPEGRVXCTKCDNGU
C HQTOWNC VJCV
U
FQGU
KPXQNXG
EQPEGRVXCTKCDNGU KP
φ Z W
9
UVTKPI QH EQPEGRV RCTCOGVGTU QDLGEV XCTKCDNGU HTQO YJKEJ
φ Z
ψ Z
WX9
XCNWGU
X
U
YJGTG
9 K
W9
9 K
U
CPF TGRNCEG GCEJ QEEWTTGPEG QH
XK
U
9 K
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ECP DG TGEQXGTGF D[ UWDUVKVWVKPI
9 K
VQ QDVCKP C HQTOWNC
9 K
HQT
XK
WK
CPF
9
HQT C
6CMG OQTG
ψ Z
WX9
6JGP VGNNU WU
XCNWGU QH VJG RCTCOGVGTU CPF UKPEG VJKU KPENWFGU VJG
CNN
W9
φ Z
XK
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ECP CRRGCT KU CVVCEJGF VQ VJG
KV HQNNQYU VJCV
PQV
UVCPFU HQT C UVTKPI QH QDLGEV RCTCOGVGTU
VJG QPN[ YC[
FGVGTOKPGU C EQPEGRV HQT
W
U
GCEJ KPUVCPEG QH EQORTGJGPUKQP HQT
HQNNQYU HTQO CP KPUVCPEG VJCV FQGU
PQV KPXQNXKPI
FGVGTOKPGU C EQPEGRV 6JWU YG JCXG VJG HQNNQYKPI
UVTGPIVJGPGF XGTUKQP QH
∃: ∀Z : Z ↔ φ Z YJGTG φ Z KU C HQTOWNC QH .
VJCV
GKVJGT QT
EQPVCKPU PQ DQWPF EQPEGRVXCTKCDNGU :
U
FQGU PQV EQPVCKP VJG U[ODQN
;
<
W
6JKU UVTGPIVJGPGF XGTUKQP KPENWFGU VJG EQORTGJGPUKQP UEJGOG HQT 28 YJKEJ CNNQYU PQ DQWPF EQPEGRV XCTKCDNGU 6JG KPUVCPEG QH EQORTGJGPUKQP VJCV IQV (TGIG KPVQ VTQWDNG KV YKNN DG TGECNNGF YCU HQT
∃: Z U: `:Z 6JKU VTQWDNGUQOG KPUVCPEG KPXQNXGU DQVJ C DQWPF EQPEGRVXCTKCDNG CPF CP QEEWTTGPEG QH U +PUKUVKPI VJCV VJG UCOG HQTOWNC OC[ PQV KPXQNXG DQVJ VWTPU QWV VQ DG GPQWIJ VQ DNQEM VJG RCTCFQZ +V KU EQPXGPKGPV VQ VJKPM QH UGEQPFTQWPF EQPEGRVXCTKCDNGU
:
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8 CU DGKPI QDVCKPGF KP VYQ OQTG UVCIGU (KTUV NGV
;
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W CPF NGV 6
.
CFF
CFF EQORTGJGPUKQP
∃: ∀Z : Z ↔ φ Z YJGTG φ Z KU C HQTOWNC QH .
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C
D
U U
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.
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CFF VJG GZVGPUKQP U[ODQN
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\
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CPF NGV
6
℘
8 CFF VJG VYQ PGY HQTOU .CY 8
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9G IGV C UVTGPIVJGPGF XGTUKQP QH TGNCVGF VQ KV CU KU TGNCVGF VQ CPF VJKU UVTGPIVJGPGF XGTUKQP KPENWFGU VJG EQORTGJGPUKQP UEJGOG HQT 28 # OQFGN HQT UGV CPF VCMG
/
℘
8 YKNN DG QDVCKPGF KP HQWT UVCIGU (KTUV NGV
/
DG CP[ EQWPVCDN[ KPHKPKVG
CU VJG FQOCKP HQT VJG QDLGEV XCTKCDNGU +H YG VCMG VJG HWNN RQYGT UGV
℘ / QT UGV
QH CNN UWDUGVU QH
/
CU VJG FQOCKP HQT VJG HKTUVTQWPF EQPEGRV XCTKCDNGU YG YKNN IGV C OQFGN QH VJG
U ℘8 (QT CZKQO YKNN JQNF KH CPF QPN[ KH VJG FGPQVCVKQP QH U
EQORTGJGPUKQP CZKQOU $WV VJGTG KU PQ YC[ YKVJ YKNN IKXG WU C OQFGN QH
6
OQFGN VQ CUUKIP C FGPQVCVKQP VQ
VJKU
VJCV OCRU
VJG FQOCKP QH VJG HKTUVTQWPF EQPEGRV XCTKCDNGU QPGVQQPG KPVQ VJG FQOCKP QH VJG QDLGEV XCTKCDNGU
℘ / ECPPQV DG OCRRGF QPGVQQPG KPVQ / D[ %CPVQTOU VJGQTGO
CPF
5Q KPUVGCF YG CRRN[ VJG .ÑYGPJGKO5MQNGO VJGQTGO YJKEJ CRRNKGF VQ VJKU UKVWCVKQP VGNNU WU VJCV VJGTG KU C EQWPVCDNG UWDUGV HKTUVTQWPF EQPEGRV XCTKCDNGU
UVKNN
/
QH
℘ / UWEJ VJCV VCMKPI /
IKXGU C OQFGN QH
6
5KPEG
/
CU VJG FQOCKP QH VJG
CPF
CTG DQVJ EQWPVCDN[
/
KPHKPKVG VJGTG YKNN DG C QPGVQQPG HWPEVKQP HTQO VJG HQTOGT KPVQ VJG NCVVGT 9KVJ CP G[G VQ HWTVJGT FGXGNQROGPVU EJQQUG UWEJ C QPGVQQPG HWPEVKQP PQV
KP VJG TCPIG QH
A
A
HQT YJKEJ KPHKPKVGN[ OCP[ GNGOGPVU QH
(QT KPUVCPEG HKZ GPWOGTCVKQPU QH
/
CPF
/
/
CPF OCR VJG HKTUV UGEQPF
W GNGOGPVU QH VJG HQTOGT VQ VJG HKTUV VJKTF HKHVJ W GNGOGPVU QH VJG NCVVGT 6CMKPI A FGPQVCVKQP QH U VJGP YKNN UWHHKEG VQ IKXG WU C OQFGN QH 6 ℘8 VJKTF
UGEQPFTQWPF EQPEGRV XCTKCDNGU VJG HWNN RQYGT UGV
6
CU VJG
9G PQY TGRGCV VJG RTQEGUU 6Q IGV C OQFGN QH
CTG
HKTUV VCMG CU VJG FQOCKP QH VJG
℘ / 1H EQWTUG VJKU OGCPU VJCV VJG UGVU KP /
YKNN DG FQKPI FQWDNGFWV[ CU RQUUKDNG XCNWGU QH HKTUVTQWPF EQPEGRV XCTKCDNGU CPF CU RQUUKDNG XCNWGU QH UGEQPFTQWPF EQPEGRV XCTKCDNGU DWV VJCV FQGU PQV OCVVGT 0QY CRRN[ VJG .ÑYGPJGKO5MQNGO VJGQTGO CICKP VQ IGV C EQWPVCDNG UWDUGV
/
QH
VJCV VCMGP CU VJG FQOCKP QH VJG UGEQPFTQWPF EQPEGRV XCTKCDNGU UVKNN IKXGU C OQFGN QH
6
VJGTG CTG KPHKPKVGN[ OCP[ GNGOGPVU QH EQTTGURQPFGPEG
U
I
DGVYGGP
/
/
VJCV CTG
KP VJG TCPIG QH
PQV
CPF UWEJ GNGOGPVU QH
DGECWUG C TGSWKTGU CP[ GNGOGPV QH
/
/
U
VJG HWPEVKQP
IKXGU C OQFGN QH
A
6
VJCV UGPFU CP GNGOGPV
℘
#
QH
/
PGKVJGT QH KP
A
PQT QH
A
5KPEG
VJGTG YKNN DG C QPGVQQPG
9G ECPPQV LWUV VCMG /
I
CU VJG FGPQVCVKQP QH
VQ DG CUUKIPGF VJG
U CPF U 5Q KPUVGCF VCMG CU VJG FGPQVCVKQP VQ A # KH # KU KP / CPF VQ I # KH PQV 6JKU
8
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℘ /
VJCV YCU CNTGCF[ RTGUGPV KP
UCOG QDLGEV D[ VJG HWPEVKQPU UGTXKPI CU FGPQVCVKQPU QH QH
A
PCOGN[ VJG XCNWGU
I #
HQT VJG KPHKPKVGN[ OCP[
#
/
VJCV CTG KP VJG TCPIG
KP
/
VJCV YGTG CNTGCF[
6JG RTQEGUU ECP DG KVGTCVGF KPFGHKPKVGN[ CPF CRRNKGU VQ RQN[CFKE CU YGNN CU OQPCFKE
/
UGEQPFQTFGT NQIKE CPF KU CFCRVCDNG VQ VJGQTKGU YKVJ *WOGOU RTKPEKRNG KP RNCEG QH .CY 8 UQ YG JCXG VJG HQNNQYKPI TGUWNV
4GUVTKEVGF *GEM %QPUKUVGPE[ 6JGQTGO
4COKHKGF RTGFKECVKXG UGEQPFQTFGT NQIKE YKVJ .CY 8
QT *WOGOU RTKPEKRNG CU CP CZKQO KU EQPUKUVGPV
6JG SWCNKHKGT
Q4GUVTKEVGFR CRRGCTU KP VJG PCOG QH VJG VJGQTGO DGECWUG VJG U[UVGOU HQT YJKEJ
*GEM RTQXGF EQPUKUVGPE[ CTG
UVTQPIGT
VJCP VJQUG KP VJG 28UGTKGU CPF JGPEG JKU EQPUKUVGPE[
TGUWNV KU UVTQPIGT VJCP YJCV JCU LWUV DGGP UVCVGF /QTG YKNN DG UCKF QP VJKU VQRKE VQYCTFU VJG GPF QH VJKU EJCRVGT
%QPUKUVGPE[ 2TQQH6JGQTGVKE2TQQHU
.GV
.
DG VJG NCPIWCIG YKVJ QPN[ QDLGEV XCTKCDNGU
Z [ \
W CPF YKVJ PQ PQPNQIKECN
TGNCVKQP QT HWPEVKQPU[ODQNU DWV LWUV VJG KFGPVKV[ U[ODQN .GV CTG LWUV VJG HQTOWNCU YG MPQY VJCV
6
6
DG VJG VJGQT[ YJQUG CZKQOU
∃MZ Z Z CUUGTVKPI VJCV VJGTG GZKUV CV NGCUV M QDLGEVU HQT CNN M n *QY FQ
KU EQPUKUVGPV VJCV PQ EQPVTCFKEVKQP ECP DG FGFWEGF HTQO KV!
Q#P[ KPHKPKVG UGV HQT KPUVCPEG VJG UGV ] W_ QH CNN PCVWTCN PWODGTU YKNN DG C OQFGN QH 6 R NGCXKPI VCEKV QCPF UQ D[ VJG UQWPFPGUU VJGQTGO 6 KU EQPUKUVGPVR 6JG RTQQH VJGQTKUV OKIJV UC[ Q$GECWUG CP[ FGFWEVKQP QH C EQPVTCFKEVKQP HTQO 6 6JG OQFGN VJGQTKUV YKNN UC[
YQWNF WUG QPN[ HKPKVGN[ OCP[ QH KVU CZKQOU $WV HQT GCEJ OQFGN ]
P
YG ECP GZRNKEKVN[ GZJKDKV C HKPKVG
W P_ QH CNN VJG ∃MZ Z Z HQT M m PR *GTG YG JCXG KP OKPKCVWTG VJG FKHHGTGPEG
QKPHKPKVKUVKER OGVJQFU QH OQFGN VJGQT[ YJKEJ TGUKUV HQTOCNK\CVKQP KP CP[ QH VJG CTKVJOGVKECN VJGQTKGU FKUEWUUGF KP QWT UWTXG[ KP VJG RTGEGFKPI EJCRVGT CPF VJG QHKPKVKUVKER OGVJQFU
DGVYGGP VJG
QH RTQQHVJGQT[ YJKEJ IGPGTCNN[ FQ PQV TGSWKTG HQT VJGKT HQTOCNK\CVKQP CP[VJKPI DG[QPF 3M HQT M
QT PQV OWEJ NCTIGT CPF KP VJKU KPUVCPEG FQ PQV GXGP TGSWKTG UQ OWEJ +P VJKU UGEVKQP + YKNN HKTUV WUG VJTGG RTKPEKRNGU MPQYP VQ JCXG HKPKVKUVKE RTQQHU VQ GUVCDNKUJ C
PWODGT QH NGOOCU YJKEJ NGOOCU YKNN VJGP DG WUGF VQ IKXG RTQQHVJGQTGVKE HKPKVKUVKE RTQQHU QH VJG EQPUKUVGPE[ QH 28 28 CPF 2*2 + JCXG CNTGCF[ GZRNCKPGF JQY VJG GZKUVGPEG QH UWEJ RTQQHU KORNKGU D[ EQPUKFGTCVKQPU TGNCVGF VQ )ÑFGNOU UGEQPF KPEQORNGVGPGUU VJGQTGO VJCV YG CTG PQV IQKPI VQ DG CDNG VQ IQ HCT WR VJG UECNG QH VJGQTKGU 3 3 3
W KP KPVGTRTGVKPI YGCM CTKVJOGVKEU
KP RTGFKECVKXG (TGIGCP VJGQTKGU 6JG VJTGG RTKPEKRNGU VJCV YKNN DG VJG OCKP KPITGFKGPVU QH VJG YQTM KP VJKU UGEVKQP CTG 5JQGPHKGNFOU VJGQTGO
VJGQTKGU VJG
YJKEJ YG OGV OQUV TGEGPVN[ KP VJG FKUEWUUKQP QH \KI\CI QT 9CPIUV[NG UGV
.ÑYGPJGKO$GJOCPP VJGQTGO
YJKEJ YG OGV GCTN[ KP VJKU EJCRVGT YJGP FKUEWUUKPI
JQY OWEJ UGV VJGQT[ YG IGV HTQO UKORNG RTGFKECVKXG UGEQPFQTFGT NQIKE RNWU .CY 8 CPF
%TCKIOU
ψ KU NQIKECNN[ FGFWEKDNG HTQO φ VJGP VJGTG KU C HQTOWNC θ ECNNGF CP KPVGTRQNCPV UWEJ VJCV ψ KU NQIKECNN[ FGFWEKDNG HTQO θ KPVGTRQNCVKQP VJGQTGO
CPF
HQT HKTUVQTFGT NQIKE %TCKIOU VJGQTGO VGNNU WU VJCV KH
θ KU NQIKECNN[ FGFWEKDNG HTQO φ CPF θ KPXQNXGU QPN[ XQECDWNCT[ EQOOQP VQ φ CPF ψ YJKEJ KU
θ KPXQNXGU PQ PQPNQIKECN U[ODQNU QVJGT VJCP QPGU VJCV QEEWT DQVJ KP φ CPF KP ψ 6GZVDQQM RTGUGPVCVKQPU QH VJKU CPF VJG QVJGT VYQ TGUWNVU M YJGP VGZVDQQMU RTGUGPV VJGO CV CNN M VQ UC[ VJCV
VGPF VQ IKXG KPHKPKVKUVKE OQFGNVJGQTGVKE RTQQHU HQT VJGO DGECWUG VJGUG CTG VJG GCUKGUV RTQQHU VQ IKXG $WV KP GCEJ ECUG VJGTG KU CP JKUVQTKECNN[ GCTN[ VJQWIJ OQTG FKHHKEWNV HKPKVKUVKE RTQQH
6JG %TCKI CPF .ÑYGPJGKO$GJOCPP VJGQTGOU CTG WUGF VQIGVJGT VQ IKXG C HKPKVKUVKE RTQQH VJGQTGVKE RTQQH QH YJCV + YKNN ECNN VJG
URNKVVKPI NGOOC
CPF VJG
GSWKPWOGTCVKQP NGOOC
YJKEJ +
OWUV PQY UVCVG #NN VJGQTKGU KP VJKU UGTKGU QH NGOOCU CTG HKTUVQTFGT
.GV GCEJ NGV
M
6
DG C VJGQT[ KP C NCPIWCIG
.
%CNN C HQTOWNC
φ Z QH . C URNKVVKPI HQTOWNC HQT 6 KH HQT
n VJG HQTOWNC ∃Mφ M KU C VJGQTGO QH 6 .GV . CFF C QPGRNCEG TGNCVKQP U[ODQN 2
CPF
CFF VJG CZKQOU
`∃Z 2Z `φ Z
∃MZ 2
6JGP
6
Z
6
∃M `2
KU ECNNGF VJG
Z
φ Z HQT GCEJ M n
URNKVVKPI GZVGPUKQP
QH
6
HQT
φ
6JG
URNKVVKPI NGOOC
UC[U VJCV C URNKVVKPI
GZVGPUKQP QH C EQPUKUVGPV VJGQT[ KU EQPUKUVGPV /QFGNVJGQTGVKECNN[ VJKU KU VTKXKCN 5KPEG CZKQOU
6
KU EQPUKUVGPV KV JCU C OQFGN HQT YJKEJ D[ VJG
∃Mφ M VJG UGV QH QDLGEVU UCVKUH[KPI φ Z OWUV DG KPHKPKVG #P[ KPHKPKVG UGV OC[ DG FKXKFGF
KPVQ C RCKT QH FKULQKPV KPHKPKVG UWDUGVU CPF VCMKPI QPG QH VJG RCKT CU VJG FGPQVCVKQP QH 2 YG IGV C OQFGN QH VJG CZKQOU CPF QH
6
2TQQHVJGQTGVKECNN[ QPG RTQEGGFU CU HQNNQYU 9G OC[ YKVJQWV NQUU QH IGPGTCNKV[ CUUWOG
φ Z JCU VJG HQTO 2Z HQT UQOG QPGRNCEG TGNCVKQP U[ODQN QH . UKPEG QVJGTYKUG YG ECP TGRNCEG . D[ VJG NCPIWCIG LWUV NKMG KV GZEGRV HQT CFFKPI C 2 CDDTGXKCVKQP φ (QT UJQTV NGV WU YTKVG ∃M2 HQT
VJCV
∃MZ2Z CPF UKOKNCTN[ KP QVJGT UKORNG ECUGU 6JWU VJG CZKQO OC[ DG YTKVVGP `∃ 2
`2 CPF VJG
CZKQO OC[ DG YTKVVGP NGOOC
6
∃M2
∃M `2
2 0QY UWRRQUG VJCV EQPVTCT[ VQ VJG EQPENWUKQP QH VJG
KU KPEQPUKUVGPV 6JGP C EQPVTCFKEVKQP ECP DG NQIKECNN[ FGFWEGF HTQO UQOG HKPKVGN[
OCP[ QH KVU CZKQOU 5KPEG
∃M2
KU FGFWEKDNG HTQO
∃O2
n M CPF UKOKNCTN[ HQT `2
HQT
O
2 YG
O
9G VJGP IGV C EQPVTCFKEVKQP HTQO VJG
OC[ CUUWOG
∃O2
EQPLWPEVKQP
τ QH UQOG HKPKVGN[ OCP[ CZKQOU QH 6 VQIGVJGT YKVJ VJG HQNNQYKPI HQTOWNC ψO
CPF
`∃ 2`2
∃O `2
∃O2
2 CTG WUGF HQT LWUV QPG
∃O `2
'SWKXCNGPVN[ `ψO KU FGFWEKDNG HTQO
2
τ 6JG %TCKI KPVGTRQNCVKQP VJGQTGO KORNKGU VJCV VJGTG KU CP
θ UWEJ VJCV QPG VJG QPG JCPF θ KU NQIKECNN[ FGFWEKDNG HTQO τ CPF JGPEG HTQO 6 YJKNG QP VJG QVJGT JCPF `ψO KU NQIKECNN[ FGFWEKDNG HTQO θ /QTGQXGT VJG KPVGTRQNCPV EQPVCKPU PQ
KPVGTRQNCPV
TGNCVKQP QT HWPEVKQP U[ODQNU QVJGT VJCP VJG NQIKECN QPG CPF QPGU VJCV CTG CRRGCT KP DQVJ
τ CPF
`ψO #PF VJG QPN[ U[ODQN QH VJG NCVVGT MKPF KU 2 .GV
P
XCTKCDNGU KP
DG CP[ PWODGT ITGCVGT VJCP QT GSWCN VQ VJG OCZKOWO QH
O
CPF VJG PWODGT QH DQWPF
θ 5KPEG ψO KU FGFWEKDNG HTQO ψP KV HQNNQYU VJCV `ψP KU FGFWEKDNG HTQO θ $[ VJG
.ÑYGPJGKO$GJOCPP VJGQTGO
θ KU NQIKECNN[ GSWKXCNGPV VQ C FKULWPEVKQP QH EQPLWPEVKQPU GCEJ
EQPUKUVKPI QH VYQ EQPLWPEVU VJG HKTUV DGKPI GKVJGT DGKPI GKVJGT
∃M2 HQT UQOG M P QT ∃P2 CPF VJG UGEQPF
∃M`2 HQT UQOG M P QT ∃P`2 5KPEG 6 KU EQPUKUVGPV θ KU FGFWEKDNG HTQO 6 CPF ∃P2
CPF VJGTGHQTG `∃M2 KU FGFWEKDNG HTQO
6
∃P2 CU KVU HKTUV
VJGTG OWUV DG CV NGCUV QPG FKULWPEV YKVJ
θ KU NQIKECNN[ FGFWEKDNG HTQO CP[ QH KVU FKULWPEVU CPF UKPEG ∃P`2 KU NQIKECNN[ FGFWEKDNG HTQO ∃P`2 VJGTG KU UQOG M m P UWEJ VJCV `ψP KU NQIKECNN[ FGFWEKDNG HTQO ∃P2
EQPLWPEV 5KPEG
∃M`2 #PF UKPEG VJG VJKTF EQPLWPEV QH ψP KORNKGU ∃P2 CEVWCNN[ `ψP KU NQIKECNN[ FGFWEKDNG HTQO ∃M`2 CNQPG $WV VJKU KU CDUWTF UKPEG HQT CP[ P CPF M YG ECP GZRNKEKVN[ GZJKDKV C OQFGN QH ∃M`2 ψP YKVJ P M QDLGEVU QH YJKEJ P UCVKUH[ 2 2Z `2Z CPF
M
Z
2Z
UCVKUH[ `2Z `2Z 6JG UWRRQUKVKQP VJCV
P
6
UCVKUH[ `2Z 2Z PQPG UCVKUH[
KU KPEQPUKUVGPV JCU DGGP TGFWEGF VQ
CDUWTFKV[
φ Z CPF ψ Z CTG ECNNGF GSWKPWOGTCVKQP HQTOWNCU HQT 6 KH `∃ φψ KU FGFWEKDNG HTQO 6 CPF ∃Mφ CPF ∃Mψ CTG FGFWEKDNG HTQO 6 HQT GCEJ M #ICKP NGV
0QVG VJCV H
CPF NGV
6
DG C VJGQT[ KP C NCPIWCIG
6
CFF CU CP CZKQO VJG EQPLWPEVKQP
∀[ ψ [ → ∃Z φ Z H Z [
5Q VJG
(QTOWNCU
∃M φ`ψ CPF ∃M `φψ HQNNQY KOOGFKCVGN[ .GV . CFF C QPGRNCEG HWPEVKQP U[ODQN
∀Z φ Z → ψ H Z ∀Z ∀Z φ Z φ Z → H Z
.
H Z
χ φ ψ QH VJG HQNNQYKPI
↔Z
Z
χ φ ψ UC[U VJCV H GUVCDNKUJGU C QPGVQQPG EQTTGURQPFGPEG DGVYGGP VJG Z UWEJ VJCV φ Z CPF Z
UWEJ VJCV
ψ Z 6JKU 6 KU ECNNGF VJG GSWKPWOGTCVKQP GZVGPUKQP QH 6 HQT φ CPF ψ 6JG
GSWKPWOGTCVKQP NGOOC
UC[U VJCV CP GSWKPWOGTCVKQP GZVGPUKQP QH C EQPUKUVGPV VJGQT[ KU EQPUKUVGPV
/QFGNVJGQTGVKECNN[ VJKU KU CICKP CP GCU[ EQPUGSWGPEG QH VJG .ÑYGPJGKO5MQNGO VJGQTGO 2TQQHVJGQTGVKECNN[ QPG DGIKPU D[ CUUWOKPI
φ CPF ψ CTG QPGRNCEG RTGFKECVGU 2
CPF CRRN[KPI VJG %TCKI KPVGTRQNCVKQP VJGQTGO VQ IGV C
θ UWEJ VJCV θ KU FGFWEKDNG HTQO 6
`χ 2 2 KU FGFWEKDNG HTQO VJGQTGO KORNKGU VJCV
θ CPF θ KPXQNXGU QPN[ CPF 2
∃P 2
θ OC[ DG VCMGP VQ DG C FKULWPEVKQP QH EQPLWPEVKQPU QH UVCVGOGPVU CDQWV VJG `2 CPF
∃P `2
2 KV HQNNQYU VJCV HQT UQOG
HQTOWNCU LWUV OGPVKQPGF VQIGVJGT YKVJ
∃M `2
P
P
UCVKUH[KPI 2`2
KU VJG KFGPVKV[ QP VJGUG NCUV
M
P
CPF
`2 CTG KPEQPUKUVGPV YKVJ
CDUWTF UKPEG KV KU RQUUKDNG VQ GZJKDKV GZRNKEKVN[ C HKPKVG OQFGN YKVJ P 22
CPF 2
CPF 2 6JG .ÑYGPJGKO$GJOCPP
PWODGTU QH QDLGEVU UCVKUH[KPI GCEJ QH VJG HQWT EQODKPCVKQPU QH 2 CPF 2 7UKPI VJG HCEV `∃ 22 CPF
UCVKUH[KPI `22 CPF
M
M
M
6
KORNKGU
m P VJG VJTGG
χ 2
2 $WV VJKU KU
QDLGEVU PQPG UCVKUH[KPI
UCVKUH[KPI `2`2 YKVJ C HWPEVKQP VJCV
QDLGEVU CPF UYKVEJGU VJG VYQ ENCUUGU QH
P
QDLGEVU
6JGUG VYQ DCUKE NGOOCU KORN[ C PWODGT QH QVJGTU #ICKP NGV .
.GV
6
DG C VJGQT[ KP C NCPIWCIG
φ Z DG C URNKVVKPI HQTOWNC HQT 6 .GV . CFF C QPGRNCEG TGNCVKQP U[ODQN H CPF NGV 6 CFF CU
CP CZKQO VJG EQPLWPEVKQP QH VJG HQNNQYKPI
∀Zφ HZ ∀Z∀[ H Z H [ ↔ Z [ ∃M[ φ [ `∃Z H Z [ HQT CNN M n
φ JQNFU YKVJ KPHKPKVGN[ OCP[ UWEJ QDLGEVU NGHV QXGT 6JKU 6 KU ECNNGF VJG KPLGEVKQP GZVGPUKQP QH 6 HQT φ 6JG 5Q
6
UC[U
CNN
QDLGEVU ECP DG OCRRGF QPGVQQPG KPVQ VJG QDLGEVU HQT YJKEJ
KPLGEVKQP NGOOC
UC[U VJCV CP KPLGEVKQP GZVGPUKQP QH C EQPUKUVGPV VJGQT[ KU EQPUKUVGPV
6Q UGG VJKU YG OC[ CUUWOG
φ KU C RTGFKECVG 2 #RRN[ URNKVVKPI VQ 2 VQ IGV 2
VQ 2 VQ IGV 2 #RRN[ GSWKPWOGTQUKV[ VYKEG VQ IGV C QPGVQQPG EQTTGURQPFGPEG CPF 2 CPF C QPGVQQPG EQTTGURQPFGPEG WPKSWG
[
UWEJ VJCV GKVJGT `2Z CPF
[
J
I Z
CPF URNKVVKPI
I
DGVYGGP `2
DGVYGGP 2 CPF 2`2 &GHKPG H Z VQ DG VJG QT 2Z CPF
[
J Z
CPF YG CTG FQPG
4GECNNKPI VJCV OCP[UQTVGF VJGQTKGU CTG TGFWEKDNG VQ QPGUQTVGF VJGQTKGU D[ KPVTQFWEKPI CRRTQRTKCVG QPGRNCEG TGNCVKQP U[ODQNU CNN VJTGG NGOOCU OGPVKQPGF UQ HCT JCXG OCP[UQTVGF XGTUKQPU #ICKP NGV
6
DG C EQPUKUVGPV KP C NCPIWCIG
.
.GV
φ Z DG C URNKVVKPI HQTOWNC HQT 6 CPF NGV
ψ Z CPF θ Z [ DG HQTOWNCU UWEJ VJCV KV KU FGFWEKDNG HTQO 6 VJCV ∃Zψ Z CPF VJCV θ FGHKPGU CP GSWKXCNGPEG TGNCVKXG VQ
ψ +P QVJGT YQTFU 6 JCU VJG HQNNQYKPI CU VJGQTGOU
∀Z ψ Z → θ Z Z
∀Z∀[ ψ Z ψ [ → θ Z [ → θ [ Z
∀Z∀[∀\ ψ Z ψ [ ψ \ → θ Z [ θ [ \ → θ Z \
.GV
DG CU KP VJG KPLGEVKQP NGOOC CPF NGV
.
6
CFF CU CZKQOU VJG HQNNQYKPI
∀Z ψ Z → φ H Z ∀Z∀[ ψ Z ψ [ → H Z H [ ↔ θ Z [
CPF CU CDQXG HQT CNN
M
n 5Q 6 UC[U VJCV H OCRU VJG QDLGEVU HQT YJKEJ ψ JQNFU VQ VJQUG HQT
φ JQNFU UGPFKPI GSWKXCNGPV CPF QPN[ GSWKXCNGPV CTIWOGPVU VQ VJG UCOG XCNWGU YKVJ KPHKPKVGN[ OCP[ QDLGEVU HQT YJKEJ φ JQNFU NGHV QXGT 6JKU 6 KU ECNNGF VJG TGRTGUGPVCVKXG GZVGPUKQP YJKEJ
QH
6
HQT
φ CPF ψ CPF θ 6JG TGRTGUGPVCVKXGU NGOOC UC[U VJCV C TGRTGUGPVCVKXG GZVGPUKQP QH C
EQPUKUVGPV VJGQT[ KU EQPUKUVGPV
9G OC[ UWRRQUG
.
JCU CV NGCUV QPG EQPUVCPV
C
DG EQPUKUVGPVN[ KP HCEV EQPUGTXCVKXGN[ CFFGF UKPEG
6
HQT YJKEJ RTQXGU
6
RTQXGU
ψ C HQT KH PQV QPG ECP
∃Zψ Z 9G OC[ VJGP KPVTQFWEG CP
YJKEJ CFFU CP GZVTC UQTV ξ υ ζ W QH XCTKCDNGU CPF PGY QPGRNCEG HWPEVKQP U[ODQN T VCMKPI CTIWOGPVU QH UQTV CPF IKXKPI XCNWGU QH UQTV ξ CPF CP CWZKNKCT[ VJGQT[ T YJKEJ CFFU VJG HQNNQYKPI CU CZKQOU CWZKNKCT[ NCPIWCIG
.
T
Z
H
6
∀Z `ψ Z → HT Z HT C
∀Z∀[ ψ Z ψ [ → HT Z HT [ ↔ θ Z [
+V KU HCKTN[ GCUKN[ UGGP VJCV T KU EQPUKUVGPV /QFGNVJGQTGVKECNN[ CU VJG FQOCKP QH VJG PGY UQTV QH XCTKCDNGU YG OC[ VCMG GSWKXCNGPEG ENCUUGU QH QDLGEVU HQT YJKEJ ψ JQNFU WPFGT VJG GSWKXCNGPEG IKXGP D[ θ YKVJ T FGPQVKPI VJG HWPEVKQP VCMKPI GCEJ QDLGEV HQT YJKEJ ψ JQNFU VQ KVU UGV QH GSWKXCNGPVU CPF VQ CNN QVJGTU VJG UGV QH GSWKXCNGPVU QH VJG QDLGEV FGPQVGF D[ 2TQQHVJGQTGVKECNN[ VJGTG KU C FKTGEV KPVGTRTGVCVKQP QH T KP CUUQEKCVKPI CP XCTKCDNG VQ GCEJ ξ XCTKCDNG KPVGTRTGVKPI ∀ξ W CU ∀ ψ → W KPVGTRTGVKPI T
CU VJG WPKSWG UWEJ VJCV GKVJGT `ψ CPF QT ψ CPF CPF KPVGTRTGVKPI ξ υ CU θ YJGTG CPF CTG VJG XCTKCDNGU CUUQEKCVGF VQ VJG ξXCTKCDNGU ξ CPF υ .GV TT CFF C QPGRNCEG HWPEVKQP U[ODQN TT VCMKPI CTIWOGPVU QH UQTV ξ CPF IKXKPI XCNWGU QH UQTV CPF NGV TT CFF VJG HQNNQYKPI CZKQOU 6
H
C
6
Z
Z
[
6
Z
H
Z
Z
Z
Z
[
[
.
Z
Z
Z
[
[
C
Z
H
6
∀ξφ HTT ξ
∀ξ∀υ HTT ξ HTT υ ↔ ξ W ∃M[ φ [ `∃ξ HTT ξ [ HQT CNN M
n
6JGP TT KU EQPUKUVGPV D[ CP CRRTQRTKCVG OCP[UQTVGF XGTUKQP QH VJG KPLGEVKQP NGOOC 6JGP ECP DG KPVGTRTGVGF KP TT UKORN[ D[ FGHKPKPI CU TT T # HQNF KU C RNCEG TGNCVKQP YKVJ VJG RTQRGTVKGU CPCNQIQWU VQ VJQUG QH CP GSWKXCNGPEG (QT GZCORNG C VTKRNG GSWKXCNGPEG YQWNF DG C RNCEG TGNCVKQP YKVJ VJG HQNNQYKPI RTQRGTVKGU 6
6
6
P
H Z
GSWKXCNGPEG
P
H
H
Z
'
4Z Z Z Z Z Z
4Z Z Z [ [ [
→ 4[[[ZZZ
[
4Z Z Z [ [
→
4[ [ [ \ \ \
4Z Z Z \ \ \
*GTG CTG LWUV TGHNGZKXKV[ U[OOGVT[ CPF VTCPUKVKXKV[ KP VTKRNKECVG 6JG TGRTGUGPVCVKXGU NGOOC JQNFU CNUQ HQT HQNF GSWKXCNGPEGU /QFGNVJGQTGVKECNN[ VJG FQOCKP QH VJG CWZKNKCT[ XCTKCDNGU ξ YQWNF DG VJG UGV QH UGVU QH GSWKXCNGPVU QH VWRNGU WPFGT VJG GSWKXCNGPEG QP P
P
VWRNGU QH QDLGEVU CUUQEKCVGF KP VJG QDXKQWU YC[ YKVJ VJG HQNF GSWKXCNGPEG QP QDLGEVU 2TQQH VJGQTGVKECNN[ VJG KPVGTRTGVCVKQP YQWNF DG QPG KP YJKEJ C UKPING SWCPVKHKGT ∀ξ KU TGRNCEGF D[ C VTKRNG QH SWCPVKHKGTU ∀ ∀ ∀ CPF VJGP CRRTQRTKCVGN[ TGNCVKXK\KPI 6JKU KU C OQTG IGPGTCN PQVKQP QH KPVGTRTGVCVKQP VJCP YG JCXG EQPUKFGTGF UQ HCT (WTVJGT DGNNU CPF YJKUVNGU EQWNF DG CFFGF DWV YG JCXG PQY GPQWIJ OCEJKPGT[ HQT C PWODGT QH EQPUKUVGPE[ RTQQHU
P
P
Z
Z
Z
# HKPKVKUVKE RTQQHVJGQTGVKE RTQQH QH EQPUKUVGPE[ RTQQH HQT 28 ECP PQY DG QDVCKPGF HTQO VJG KPHKPKVKUVKE OQFGNVJGQTGVKE RTQQH QH EQPUKUVGPE[ HQT ℘8 KP VJG RTGEGFKPI UGEVKQP D[ UWDUVKVWVKPI CRRGCN VQ 5JQGPHKGNFOU VJGQTGO HQT CRRGCN VQ VJG GZKUVGPEG QH VJG RQYGT UGV CPF CRRGCN VQ %TCKIOU VJGQTGO CPF VJG .ÑYGPJGKO$GJOCPP VJGQTGOU VJG NGOOCU CDQXG HQT CRRGCN VQ VJG .ÑYGPJGKO5MQNGO VJGQTGO 6Q DGIKP NGV CPF DG CU CV VJG DGIKPPKPI QH VJKU UGEVKQP 5Q CU PQ PQPNQIKECN U[ODQNU CPF JCU CU CZKQOU ∃M HQT CNN .GV CFF HKTUVTQWPF EQPEGRVXCTKCDNGU W VQ CPF NGV CPF YKVJ VJGO CVQOKE HQTOWNCU QH VJG V[RG CFF RTGFKECVKXG EQORTGJGPUKQP CZKQOU VQ 6JGP KU EQPUKUVGPV D[ 5JQGPHKGNFOU VJGQTGO .GV D[ CFF VJG U[ODQN U CPF NGV CFF VJG HQNNQYKPI CZKQO XKC
6
.
.
Z Z
6
:
;
<
Z
M
.
: Z
.
6
6
6
.
6
U
U ↔ ≡ ∃M `∃ U HQT CNN
: Z
;
:
:
;
:
Z
M
n
6JGP KU EQPUKUVGPV D[ CP CRRTQRTKCVG VYQUQTVGF XGTUKQP QH VJG KPLGEVKQP NGOOC VJG TQNGU QH ψ CPF φ DGKPI VCMGP UKORN[ D[ VJG UQTVCN RTGFKECVG HQT UQTV CPF KVU PGICVKQP TGURGEVKXGN[ 9G JCXG UGGP VJCV KPUVCPEGU QH EQORTGJGPUKQP HQT HQTOWNCU YKVJQWV DQWPF XCTKCDNGU DWV YKVJ VJG U[ODQN U HQNNQY HTQO VJG KPUVCPEGU YKVJQWV DQWPF XCTKCDNGU CPF YKVJ VJG U[ODQN U 5Q YG JCXG 28 RNWU VJG GZVTC CZKQOU 0QY YG TGRGCV VJG RTQEGUU YKVJ C UNKIJV OQFKHKECVKQP .GV CFF UGEQPFTQWPF EQPEGRV XCTKCDNGU W CPF NGV CFF RTGFKECVKXG EQORTGJGPUKQP 6JGP KU EQPUKUVGPV D[ 5JQGPHKGNFOU VJGQTGO .GV CFF C PGY HWPEVKQP U[ODQN T CPF NGV CFF VJG HQNNQYKPI CZKQOU 6
Z
:
:
.
:
;
<
6
6
.
T T T lU
:
;
:
:
6
↔ : ≡ ;
6JGP KU EQPUKUVGPV D[ CP CRRTQRTKCVG XGTUKQP QH VJG KPLGEVKQP NGOOC YKVJ VJG TQNG QH ψ DGKPI VCMGP VJKU VKOG D[ `∃ U WUKPI VJG CZKQOU CDQXG 0QY T KU QPN[ CP CWZKNKCT[ CPF PQV [GV SWKVG YJCV YG YCPV $WV KV YKNN DG RTQXCDNG KP VJCV HQT GXGT[ VJGTG GZKUVU C WPKSWG HQT YJKEJ VJG HQNNQYKPI FKULWPEVKQP JQNFU 6
:
Z
:
6
:
Z
∃ ∀ ↔ U ∨
`∃ ∀ ↔ T :
:
\ : \
\ : \
: \
: \
Z
:
Z
:
6JKU OC[ DG ECNNGF U CPF VJGP YG YKNN JCXG CNN VJG CZKQOU QH 28 6JG RTQQH HQT 2*2 2*2 KU GUUGPVKCNN[ VJG UCOG YKVJ CP CRRGCN VQ VJG TGRTGUGPVCVKXGU NGOOC QH YJKEJ VJG KPLGEVKQP NGOOC KU VJG URGEKCN ECUG YJGTG VJG GSWKXCNGPEG KPXQNXGF KU KFGPVKV[ KP RNCEG QH VJG KPLGEVKQP NGOOC $QVJ YKVJ .CY 8 CU CZKQO CPF YKVJ *2 CU CZKQO VJG RTQEGUU ECP DG KVGTCVGF KPFGHKPKVGN[ VJTQWIJ VJG UKPING FQWDNG VTKRNG RTGFKECVKXG VJGQTKGU CPF QP VQ VJGKT WPKQP VJG HWNN TCOKHKGF RTGFKECVKXG VJGQT[ #U OGPVKQPGF GCTNKGT VJG RTQQH QH )GPV\GPOU EWVGNKOKPCVKQP VJGQTGO HTQO YJKEJ CNN QWT QVJGT NGOOCU YGTG FGTKXGF KP C HCKTN[ GNGOGPVCT[ YC[ TGSWKTGU VJG UWRGTGZRQPGPVKCN HWPEVKQP CPF KU MPQYP VQ DG HQTOCNK\CDNG KP 3 +∆ UWRGTGZR UWRGTGZRQPGPVKCN HWPEVKQPCN CTKVJOGVKE +V HQNNQYU M VJQWIJ + JCXG JCTFN[ FQVVGF GXGT[ QT ETQUUGF GXGT[ KP VJG RTQQH M VJCV VJG EQPUKUVGPE[ QH 2P8 KU RTQXCDNG HQT GCEJ W KP 3 6JG RTQQH QH VJG EQPUKUVGPE[ QH VJGKT WPKQP 2ω8 YJKEJ KU VQ UC[ VJG RTQQH VJCV HQT GXGT[ VJG VJGQT[ 2P8 KU EQPUKUVGPV TGSWKTGU VJG CUUWORVKQP VJCV HQT GXGT[ VJG UWRGTGZRQPGPVKCN HWPEVKQP ECP DG KVGTCVGF VKOGU YJKEJ KU VQ UC[ TGSWKTGU VJG UWRGTFWRGTGZRQPGPVKCN QT UWRGTGZRQPGPVKCN HWPEVKQP CPF 3 +∆ UWRGTGZR UWRGTGZRQPGPVKCN HWPEVKQPCN CTKVJOGVKE 6JWU VJG TGUWNVU QH VJKU UGEVKQP OC[ DG UWOOGF WR CU HQNNQYU Z
:
K
V
P
P
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QT VYQ GCTNKGT 1P VJG UGEQPF RQKPV %JCTNGU 2CTUQPU h8+ CHVGT TGECNNKPI (TGIG U
O
FGHKPKVKQP QH VJG PWODGT QH C EQPEGRV YJKEJ 2CTUQPU YTKVGU 0Z(Z CHVGT TGECNNKPI (TGIG U
O
FGTKXCVKQP HTQO VJKU FGHKPKVKQP QH *WOG U RTKPEKRNG YJKEJ 2CTUQPU ECNNU # CPF CHVGT
O
TGECNNKPI (TGIG U FGHKPKVKQPU QH \GTQ UWEEGUUQT CPF PCVWTCN PWODGT KP VGTOU QH VJG PQVKQP QH PWODGT OCMGU VJG HQNNQYKPI TGOCTM
(TQO VJGUG VJG 2GCPQ CZKQOU ECP DG RTQXGF KV KU PQV PGEGUUCT[ VQ WUG CP[
N
CZKQOU QH UGV GZKUVGPEG GZEGRV KP KPVTQFWEKPI VGTOU QH VJG HQTO 0Z(Z
O
CPF KP RTQXKPI # UQ VJCV VJG CTIWOGPV EQWNF DG ECTTKGF QWV VCMKPI # CU CP CZKQO
9TKIJV QP VJKU RQKPV YGPV HWTVJGT VJCP 2CTUQPU KP CEVWCNN[ UMGVEJKPI C FGTKXCVKQP QH VJG 2GCPQ RQUVWNCVGU VJQWIJ VJG UMGVEJ UVQRU UJQTV QH DGKPI C HWNN[ TKIQTQWU FGFWEVKQP UWEJ C FGFWEVKQP DGKPI RTQOKUGF HQT C NCVGT LQKPV RWDNKECVKQP YKVJ 0GKN 6GPPCPV 1P VJG HKTUV RQKPV 2GVGT )GCEJ UC[U VJG HQNNQYKPI D[ YC[ QH
O
ETKVKEKUO QH C TGOCTM QH &WOOGVV U
$WV YG JCXG QP VJG EQPVTCT[ GXGT[ TGCUQP VQ DGNKGXG VJCV VJGTG KU C VQVCNKV[ ENQUGF WPFGT VJG QRGTCVKQP QH OCRRKPI CTDKVTCT[ RTGFKECVGU FGHKPGF QXGT KV QPVQ ECTFKPCN PWODGTU .GV WU EQPUKFGT VJG FQOCKP EQPUKUVKPI QH VJG PCVWTCN PWODGTU VQIGVJGT YKVJ
ℵ 6Q CP[ RTGFKECVG FGHKPGF QXGT VJKU
FQOCKP VJGTG CVVCEJGU GKVJGT C PCVWTCN PWODGT QT
ℵ CU VJG PWODGT QH
QDLGEVU QH YJKEJ VJG RTGFKECVG KU VTWG GI VJG PWODGT CVVCEJGU VQ VJG RTGFKECVG
NM
NM O
O
KU RTKOG CPF NGUU VJCP VJG PWODGT
ℵ VQ VJG RTGFKECVG
KU GXGP 9G JCXG PQ PGGF VQ IQ QWVUKFG VJG FQOCKP VQ HKPF ECTFKPCN
PWODGTU CVVCEJKPI VQ VJGUG RTGFKECVGU
9TKIJV QP VJKU RQKPV FKF PQV IQ CU HCT CU )GCEJ HQT JG FKF PQV QHHGT C OQFGN VQ UJQY *2 EQPUKUVGPV DWV OGTGN[ KPFKECVGF JQY VJG CVVGORV VQ FGTKXG VJG 4WUUGNN RCTCFQZ DTGCMU FQYP CPF EQPLGEVWTGU VJCV VJG U[UVGO YKVJ *2 TCVJGT VJCP .CY 8 KU EQPUKUVGPV 9JCV 9TKIJV FKF VJCV JCF PQV DGGP FQPG DGHQTG YCU VQ DTKPI VQIGVJGT VJG VYQ QDUGTXCVKQPU CPF QHHGT CV NGCUV CU C EQPLGEVWTG YKVJ UQOG HQTOCN UWRRQTV VJG QDUGTXCVKQP VJCV CTKVJOGVKE ECP DG FGTKXGF KP C EQPUKUVGPV VJGQT[ EQPUKUVKPI QH UGEQPFQTFGT NQIKE YKVJ *2 CU CP CFFKVKQPCN CZKQO
M
VJCV KU VQ UC[ KP VJG KORTGFKECVKXG CDUVTCEVKQPKUV VJGQT[ VJCV
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
VQ CDUVTCEVKQPKUV UGV VJGQT[ KP VJG YQTM QH $QQNQU 6JGP + YKNN UWIIGUV JQY QPG EQWNF IQ OWEJ HWTVJGT CV VJG EQUV QH DGKPI TCVJGT NGUU (TGIGCP
6Q TGVWTP VQ 9TKIJV JKU DQQM NGHV UQOG NQQUG GPFU (QT QPG VJKPI VJG LQKPV RCRGT QH 9TKIJV CPF 6GPPCPV PGXGT OCVGTKCNK\GF (QT CPQVJGT VJKPI VJG EQPUKUVGPE[ QH (# YCU UVCVGF D[ 9TKIJV OGTGN[ CU C EQPLGEVWTG CPF VJQWIJ TGUVCVGF CU HCEV VJG HQNNQYKPI [GCT D[ *CTQNF *QFGU KV YCU UVKNN NGHV YKVJQWV RTQQH 6JGUG NQQUG GPFU YGTG JQYGXGT HCKTN[ SWKEMN[ VKGF WR 1P VJG QPG JCPF C HWNN[ TKIQTQWU FGFWEVKQP QH VJG 2GCPQ RQUVWNCVGU KP (TGIG CTKVJOGVKE YCU RWDNKUJGF D[ 6GPPCPV QP JKU QYP VJQWIJ WPHQTVWPCVGN[ KP C NQEWU 6GPPCPV YJGTG KV GUECRGF VJG CVVGPVKQP QH OQUV RJKNQUQRJGTU CPF NQIKEKCPU YJQ OKIJV JCXG DGGP KPVGTGUVGF 1P VJG QVJGT JCPF VYQ TGXKGYU *C\GP CPF $WTIGUU RQKPVGF QWV VJCV EQPUKUVGPE[ HQNNQYU HTQO VJG GZKUVGPEG QH C UKORNG
O
OQFGN HQT (# VJG OQFGN YCU KPFGGF )GCEJ U DWV JKU TGOCTM JCF DGGP GPVKTGN[ QXGTNQQMGF D[ CNN EQPEGTPGF CPF TGOCKPGF UQ HQT CNOQUV VYGPV[ [GCTU WPVKN 5CWN -TKRMG ECOG CETQUU KV # HWNN[ UCVKUHCEVQT[ VTGCVOGPV JQYGXGT JCF VQ CYCKV VJG YQTM QH CPQVJGT GCTN[ TGCFGT QH 9TKIJV YJQ JCF CNUQ KPFGRGPFGPVN[ TGFKUEQXGTGF VJG )GCEJ OQFGN )GQTIG $QQNQU *G KPVTQFWEGF VJG VGTOKPQNQI[ UVKNN WUGF KP FKUEWUUKPI VJGUG OCVVGTU
RQ
RTKPEKRNG
O
RQ
(TGIG U VJGQTGO
R
Q
O
*WOG U
(TGIG CTKVJOGVKE #PF JKU DGCWVKHWN RCRGT $QQNQU
OCFG VYQ ETWEKCN PGY RQKPVU (KTUV PQV QPN[ ECP VJG 2GCPQ RQUVWNCVGU DG FGTKXGF KP (# DWV VJG[ ECP DG FGTKXGF GUUGPVKCNN[ D[ HQNNQYKPI (TGIGOU QYP EJCKP QH FGFWEVKQPU 9KVJ
O
(TGIG U FGHKPKVKQPU QH \GTQ UWEEGUUQT CPF PCVWTCN PWODGT UGEQPFQTFGT 2GCPQ CTKVJOGVKE 2 DGEQOGU KPVGTRTGVCDNG KP (# 5GEQPF JG RTQXGF VJCV (# KU PQV QPN[ EQPUKUVGPV DWV KU GSWKEQPUKUVGPV YKVJ QT QH VJG UCOG EQPUKUVGPE[ UVTGPIVJ CU 2 (QT JG RTQXGF VJG HQNNQYKPI
6JG )GCEJ$QQNQU %QPXGTUG VQ (TGIGOU 6JGQTGO
(TGIG CTKVJOGVKE KU KPVGTRTGVCDNG KP
UGEQPFQTFGT 2GCPQ CTKVJOGVKE 6JG KFGC KU VQ OKOKE VJG )GCEJ OQFGN YJKNG YQTMKPI YKVJKP UGEQPFQTFGT 2GCPQ CTKVJOGVKE YKVJ RNC[KPI VJG TQNG QH ℵ CPF [ RNC[KPI VJG TQNG QH [ 9TKVKPI 7 HQT VJG nZ Z Z CPF 〈[〉 HQT VJG UVTKEV UGIOGPV nZ Z [ YG OWUV HKTUV RTQXG VJCV GXGT[ : KU GSWKPWOGTQWU GKVJGT YKVJ 7 QT YKVJ UQOG 〈[〉 YJGTG VJGUG CNVGTPCVKXGU CTG GZENWUKXG CPF [ WPKSWG 9G ECP VJGP FGHKPG VJG QRGTCVKQP D[ NGVVKPI : DG KH : KU GSWKPWOGTQWU YKVJ 7 CPF NGVVKPI : DG [ KH : KU GSWKPWOGTQWU YKVJ 〈[〉 # DKV OQTG YKNN DG UCKF CDQWV VJG RTQQH C DGNQY
WPKXGTUCN EQPEGRV
# UGTKGU QH RCRGTU D[ $QQNQU D[ *GEM CPF QPG D[ VJG VYQ LQKPVN[ HQNNQYGF GZCOKPKPI KP UGCTEJKPI FGVCKN VJG FGTKXCVKQP QT TCVJGT UGXGTCN RQUUKDNG FGTKXCVKQPU QH VJG 2GCPQ RQUVWNCVGU CPF QVJGT TGUWNVU HTQO *2 DQVJ HTQO C NQIKECN RQKPV QH XKGY CPF HTQO CP JKUVQTKECN RQKPV QH XKGY KP EQORCTKUQP YKVJ (TGIGOU YC[ QH RTQEGGFKPI KP VJG KPHQTOCN UMGVEJ KP VJG )TWPFNCIGP CPF VJG HQTOCN VTGCVOGPV KP VJG )TWPFIGUGV\G *GTG JKUVQTKECN KUUWGU YKNN DG GPVKTGN[ NGHV CUKFG CPF CU VQ NQIKECN KUUWGU QPN[ VYQ YKNN DG CFFTGUUGF KP CP[ FGRVJ (KTUV ECP VJG CUUWORVKQPU QH (# DG UKIPKHKECPVN[ YGCMGPGF YKVJQWV NQUKPI VJG 2GCPQ RQUVWNCVGU! 5GEQPF ECP VJG CUUWORVKQPU QH (# DG UVTGPIVJGPGF KP C OCPPGT VJCV KU PQV CF JQE UQ CU VQ IKXG OQTG QH ENCUUKECN OCVJGOCVKEU! +P VJG TGOCKPFGT QH VJKU UGEVKQP + YKNN VCMG WR VJG HKTUV QH VJGUG VYQ SWGUVKQPU CPF YKNN VWTP VQ VJG UGEQPF KP VJG PGZV UGEVKQP *WOGOU RTKPEKRNG RTQXKFGU CDUVTCEVU YKVJ TGURGEV VQ VJG GSWKXCNGPEG QH GSWKPWOGTQUKV[ 5WEJ CDUVTCEVU KPENWFG VJG PCVWTCN PWODGTU YJKEJ YGTG (TGIGOU RTKOCT[ VJQWIJ D[ PQ OGCPU GZENWUKXG KPVGTGUV CU YGNN CU %CPVQTOU VTCPUHKPKVG ECTFKPCNU VJQWIJ YKVJKP (# QPG ECP QPN[ RTQXG VJG GZKUVGPEG QH VJG UOCNNGUV QH VJGUG ℵ +V YQWNF DG UWTRTKUKPI KH VJG GZKUVGPEG QH GXGP QPG VTCPUHKPKVG ECTFKPCN YGTG KPFKURGPUCDNG HQT VJG FGTKXCVKQP QH VJG 2GCPQ RQUVWNCVGU CPF KP HCEV KV KU PQV 6JG 2GCPQ RQUVWNCVGU CTG UVKNN HQTVJEQOKPI KH QPG TGUVTKEVU *2 UQ VJCV KV RTQXKFGU ECTFKPCNU QPN[ HQT HKPKVG UGVU 6JKU HCEV JCU DGGP PQVGF KP QPG HQTO QT CPQVJGT KPFGRGPFGPVN[ UGXGTCN VKOGU .GV WU UC[ VJCV C FGHKPKVKQP QH HKPKVWFG KU OKPKOCNN[ CFGSWCVG KH KV RGTOKVU VJG FGFWEVKQP QH VJG HQNNQYKPI VYQ EQPFKVKQPU PWNNKV[ CPF CFLWPEVKQP
`∃Z:Z → : HKPKVG : HKPKVG ∀Z ;Z ↔ :Z ∨ Z [ → ; HKPKVG
$QVJ CTG RTQXCDNG HQT &GFGMKPF CPF 4WUUGNN HKPKVWFG 6JGP YG JCXG VJG HQNNQYKPI CDUVTCEV XGTUKQP QH VJG TGUWNV KP SWGUVKQP 5GEQPFQTFGT 2GCPQ CTKVJOGVKE KU KPVGTRTGVCDNG KP F[CFKE UGEQPFQTFGT NQIKE YKVJ HKPKVG *2 CU CP CZKQO HQT CP[ OKPKOCNN[ CFGSWCVG PQVKQP QH HKPKVWFG 6JG 6GPPCPV*GEM$GNN 4GHKPGOGPV QH (TGIGOU 6JGQTGO
6WTPKPI HTQO YGCMGPKPIU QH *WOGOU RTKPEKRNG VQ YGCMGPKPIU QH VJG WPFGTN[KPI NQIKE C RQKPV GORJCUK\GF D[ 6GPPCPV CPF KPFGRGPFGPVN[ TGFKUEQXGTGF D[ $GNN KU VJCV VJG FGTKXCVKQP QH VJG 2GCPQ RQUVWNCVGU FQGU PQV TGSWKTG VJG NCY QH VJG GZENWFGF OKFFNG # ∨ `#
QT QVJGT RTKPEKRNGU QH ENCUUKECN NQIKE VQ YJKEJ KPVWKVKQPKUVU QDLGEV $WV KPVWKVKQPKUV XKGYU CTG DG[QPF VJG UEQRG QH VJG RTGUGPV UVWF[ /QTG TGNGXCPV QDXKQWUN[ KU VJG SWGUVKQP QH JQY HCT VJG FGTKXCVKQP QH VJG 2GCPQ RQUVWNCVGU FGRGPFU QP JCXKPI UGEQPFQTFGT NQIKE YKVJ KORTGFKECVKXG EQORTGJGPUKQP 6JKU KUUWG YCU VQWEJGF QP KP VJG RTGEGFKPI EJCRVGT DWV VJGTG KU OQTG VQ DG UCKF CDQWV VJG SWGUVKQP YJKEJ JCU GURGEKCNN[ DGGP KPXGUVKICVGF D[ «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Π HQTOWNCU QT HQTOWNCU EQPUKUVKPI QH C UVTKPI QH WPKXGTUCN UGEQPFQTFGT SWCPVKHKGTU HQNNQYGF D[ C HQTOWNC YKVJQWV HWTVJGT UGEQPFQTFGT SWCPVKHKGTU 6JGUG OC[ GSWCNN[ DG FGUETKDGF CU U[UVGOU YJGTG EQORTGJGPUKQP KU CUUWOGF HQT QPN[ HQT Σ HQTOWNCU EQPUKUVKPI QH C UVTKPI QH GZKUVGPVKCN UGEQPFQTFGT SWCPVKHKGTU HQNNQYGF D[ C HQTOWNC YKVJQWV HWTVJGT UGEQPFQTFGT SWCPVKHKGTU 6JG VYQ HQTOWNCVKQPU CTG GSWKXCNGPV DGECWUG QPEG QPG JCU C EQPEGRV RTGFKECVKXG EQORTGJGPUKQP KU GPQWIJ VQ IKXG KVU EQORNGOGPV CPF Π HQTOWNCU CPF Σ HQTOWNCU CTG GSWKXCNGPV VQ VJG PGICVKQPU QH GCEJ QVJGT #EVWCNN[ UQOG SWGUVKQPU QH HQTOWNCVKQP CTKUG CU VQ LWUV YJCV UJQWNF DG EQPUKFGTGF VJG Π XGTUKQP QH UGEQPFQTFGT 2GCPQ CTKVJOGVKE CPF YJCV UJQWNF DG EQPUKFGTGF VJG Π XGTUKQP QH (TGIG CTKVJOGVKE 1P VJG 2GCPQ UKFG YG JCXG UGGP VJCV YKVJ HWNN EQORTGJGPUKQP VJG UKPINGCZKQO XGTUKQP + QH KPFWEVKQP KU GSWKXCNGPV VQ VJG CZKQOUEJGOG XGTUKQP + DWV VJCV YJGP EQORTGJGPUKQP KU TGUVTKEVGF + KU UVTQPIGT VJCP
+ #PF KP RCTVKEWNCT VJGTG CTG VYQ Π XGTUKQPU QH UGEQPFQTFGT 2GCPQ CTKVJOGVKE VJG UVTQPIGT Π %# YKVJ + CPF VJG YGCMGT Π %# YKVJ + 6JG GZRGTKGPEG QH OCVJGOCVKECN NQIKEKCPU JCU UJQYP VJCV VJG NCVVGT U[UVGO KU VJG OQTG UKIPKHKECPV HQWPFCVKQPCNN[ +V KU VJG NCUV QH VJG (TKGFOCP U[UVGOU ( (QT RWTRQUGU QH EQORCTKUQP YKVJ (TGIG CTKVJOGVKE JQYGXGT VJG OQUV CRRTQRTKCVG U[UVGO KU C UNKIJVN[ FKHHGTGPV QPG + YKNN ECNN Π 2 HQT Π 2GCPQ CTKVJOGVKE 6JKU U[UVGO EQPUKUVU QH F[CFKE UGEQPFQTFGT NQIKE YKVJ Π EQORTGJGPUKQP CPF CZKQOU 3 CPF 3 HQT \GTQ CPF UWEEGUUQT Π 2 KU KPVGTRTGVCDNG KP Π %# YJKEJ JCU QPN[ OQPCFKE UGEQPFQTFGT NQIKE WUKPI VJG EQFKPI QH RCKTU QH PCVWTCN PWODGTU D[ UKPING PCVWTCN PWODGTU VJCV KU FGHKPCDNG KP VGTOU QH CFFKVKQP CPF OWNVKRNKECVKQP KP 2 CPF JGPEG
KP Π %# 5KPEG Π %# KU YGCMGT VJCP Π %# UQ KU Π 2 $WV Π %# CPF Π 2 CTG QH GSWKXCNGPV UVTGPIVJ DGECWUG YG JCXG VJG HQNNQYKPI &GFGMKPFOU 6JGQTGO 8GTUKQP +++
Π %# KU KPVGTRTGVCDNG KP Π 2
6JG KPVGTRTGVCVKQP ECNNGF HQT D[ VJG VJGQTGO RTQEGGFU D[ FGHKPKPI VJG PQVKQP QH KPFWEVKXG KP VJG WUWCN YC[ : KU KPFWEVKXG KH CPF QPN[ KH YG JCXG : CPF JCXG :Z YJGPGXGT :Z CPF VJGP FGHKPKPI VJG PQVKQP QH PCVWTCN PWODGT KP VJG WUWCN YC[ IKXGP VJG PQVKQP QH KPFWEVKXG Z KU C PCVWTCN PWODGT KH YG JCXG :Z HQT GXGT[ KPFWEVKXG : 6JKU FGHKPKVKQP KU GZRTGUUKDNG D[ C Π HQTOWNC ν Z YKVJ QGXGT[ KPFWEVKXG :R IKXKPI VJG KPKVKCN WPKXGTUCN UGEQPFQTFGT SWCPVKHKGT *GPEG D[ Π EQORTGJGPUKQP VJGTG GZKUVU CP UWEJ VJCV HQT CNN Z YG JCXG 0Z KH CPF QPN[ KH Z KU C PCVWTCN PWODGT #PF PQY HQT C UWDVNG DWV KORQTVCPV RQKPV +V HQNNQYU VJCV Π EQORTGJGPUKQP JQNFU HQT HQTOWNCU TGNCVKXK\GF VQ PCVWTCN PWODGTU (QT IKXGP C Π HQTOWNC φ Z UC[ QH 0
HQTO ∀:∃[∀\W KVU TGNCVKXK\CVKQP φν NQQMU NKMG VJKU
∃;∀Z X Z → ;Z ↔ ∀:∃[ ν [ ∀\ ν \ → WZW:W[W\W
6JKU KU EGTVCKPN[ PQV KVUGNH CP KPUVCPEG QH Π EQORTGJGPUKQP DGECWUG QH VJG UGEQPFQTFGT SWCPVKHKGTU KP X Z $WV KU C EQPUGSWGPEG QH VJG HQNNQYKPI VYQ KPUVCPEGU QH Π EQORTGJGPUKQP
∀7∃;∀Z ;Z ↔ 7Z ∀:∃[ 7[ ∀\ 7\ → WZW:W[W\W ∃7∀Z 7Z ↔ ν Z
# UNKIJVN[ OQTG KPHQTOCN YC[ VQ RWV VJG RQKPV YQWNF DG CU HQNNQYU 6JG HCEV VJCV ν KU C Π HQTOWNC OGCPU VJCV Π EQORTGJGPUKQP URGEKHKECNN[ VJG KPUVCPEG KORNKGU VJCV VJG EQPEGRV 0 QH DGKPI C PCVWTCN PWODGT GZKUVU QT DGNQPIU VQ VJG FQOCKP QH VJG UGEQPFQTFGT XCTKCDNGU CPF KU VJGTGHQTG CXCKNCDNG HQT WUG CU C RCTCOGVGT KP QVJGT Π HQTOWNCU VQ YJKEJ Π EQORTGJGPUKQP OC[ DG CRRNKGF CU KP VJG HQNNQYKPI
∃;∀Z ;Z ↔ 0Z ∀:∃[ 0[ ∀\ 0\ → WZW:W[W\W
# UKOKNCT RQKPV KU VJCV YG JCXG KPFWEVKQP KP VJG UKPINGCZKQO HQTO + UKPEG KH : KU KPFWEVKXG TGNCVKXG VQ PCVWTCN PWODGTU UQ VJCV YG JCXG : CPF CV NGCUV JCXG :Z YJGPGXGT
VJGP VJG KPVGTUGEVKQP QH : CPF 0 KU KPFWEVKXG VQWV EQWTV CPF GXGT[ PCVWTCN PWODGT D[ FGHKPKVKQP OWUV HCNN WPFGT VJKU KPVGTUGEVKQP CPF JGPEG WPFGT : #FFKVKQP CPF OWNVKRNKECVKQP ECP VJGP DG KPVTQFWEGF CU KP VJG RTQQH QH VJG KPVGTRTGVCDKNKV[ QH 3 KP VJG RTGEGFKPI EJCRVGT YJKEJ TGSWKTGF QPN[ RTGFKECVKXG EQORTGJGPUKQP CPF VJGP YG JCXG TGEQXGTGF VJG OQTG EQPXGPVKQPCN HQTOWNCVKQP Π
:Z CPF Z KU C PCVWTCN PWODGT
%# VJG (TKGFOCP U[UVGO ( 1P VJG (TGIG UKFG YG JCXG C EJQKEG DGVYGGP C U[UVGO JCXKPI C UKPINGCZKQO XGTUKQP QH *2 NKMG 2*2 QH VJG NCUV EJCRVGT DWV YKVJ Π EQORTGJGPUKQP TCVJGT VJCP RTGFKECVKXG EQORTGJGPUKQP CPF C U[UVGO EQORCTCDNG VQ *GEMOU YJKEJ YQWNF CNNQY VGTOU ^^Z φ Z ^^ QH CNN TCPMU CPF CP CZKQOUEJGOG XGTUKQP *2 CRRN[KPI VQ VJGO GXGP HQT HQTOWNCU φ HQT YJKEJ EQORTGJGPUKQP KU PQV CUUWOGF 6JG OQTG VTCEVCDNG U[UVGO CPF VJG QPG OQUV TGNGXCPVN[ EQORCTGF VQ Π %# CPF Π 2 KU VJG HQTOGT U[UVGO CPF KV KU VJCV U[UVGO VJCV + YKNN ECNN Π (# HQT Π (TGIG CTKVJOGVKE 9G VJGP JCXG VJG HQNNQYKPI RCKT QH TGUWNVU .KPPGDQOU 4GHKPGOGPV QH (TGIGOU 6JGQTGO
Π (TGIG CTKVJOGVKE
Π 2GCPQ CTKVJOGVKE KU KPVGTRTGVCDNG KP
%QPXGTUG VQ .KPPGDQOU 4GHKPGOGPV QH (TGIGOU 6JGQTGO 4GHKPGOGPV QH VJG )GCEJ$QQNQU %QPXGTUG VQ (TGIGOU 6JGQTGO
Π 2GCPQ CTKVJOGVKE
Π (TGIG CTKVJOGVKE KU KPVGTRTGVCDNG KP
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↔ ∃:∃; Z : : ⊂ ; [ ; Z KU RTQVQPCVWTCN ↔ ∃: ∀[ :[ ↔ [ Z Z : `:Z
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CPPQWPEGF VJCV VJGTG YGTG QDLGEVU ECNNGF
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R
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CNN
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O
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O
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:
Z
4 KP KVU HKTUV RNCEG
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TGNCVKQPCN EQPEGRV CPF C HKTUVNGXGN QPGRNCEG EQPEGRV CU CTIWOGPVU VQ UQOG QDLGEV CU
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# PQVKQP QH
UWKVCDKNKV[
YKNN DG FGHKPGF CPF VJG QVJGT CZKQOU YKNN DG HQTOWNCVGF KP
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ο KP VJCV ECUG CPF KP PQ QVJGT 6JG CZKQOU YKNN VJGP TGCF CU
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l
4 UWKVCDNG ↔ h4: ο 4 UWKVCDNG 5 UWKVCDNG h4: h5; → ∀< 4:< ↔ 5;< 4 UWKVCDNG → h4: h4; ↔ 4:;
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PGXGT
VJG UCOG VJG QVJGT VJCV VJG[ CTG VJG UCOG
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l
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O
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O
O
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*WOG U RTKPEKRNG YJKEJ YG YCPV CPF FQP V JCXG .CY 8 YJKEJ YG FQP V # NKVVNG VJQWIJV UJQYU VJCV # KORNKGU VJCV CP[ UWKVCDNG
4 KU CP GSWKXCNGPEG
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4 UVTQPIN[ UWKVCDNG KH VJG HQNNQYKPI JQNF
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FGHKPKVKQP IKXGU VJG VJKTFQTFGT
YGCM
IGPGTCN VJGQT[ QH CDUVTCEVKQPU 9(6#
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: KV KU GCUKN[ UGGP VJCV VJG HQNNQYKPI EQPFKVKQP FGHKPGU C FQWDN[ KPXCTKCPV FKEJQVQO[ 4: KPXCTKCPV
4::; ↔ :: :; ∨ `:: `:; : YQWNF IKXG TKUG VQ VJG UCOG GSWKXCNGPEG 4: +PXGTUGN[ IKXGP CP[ FQWDN[ KPXCTKCPV FKEJQVQO[ 4 KV KU GSWCNN[ GCUKN[ UGGP VJCV VJG HQNNQYKPI EQPFKVKQPU YJGTGKP 7 KU VJG WPKXGTUCN EQPEGRV FGHKPG EQORNGOGPVCT[ KPXCTKCPV : 1DXKQWUN[ VJG EQORNGOGPV QH
:4 : ↔ 47:
:4 : ↔ `47:
#UUWOKPI VJG GZKUVGPEG QH CDUVTCEVU HQT FQWDN[ KPXCTKCPV FKEJQVQOKGU KU KP GHHGEV GSWKXCNGPV VQ CUUWOKPI VJG GZKUVGPEG QH GZVGPUKQPU HQT KPXCTKCPV UGEQPFNGXGN EQPEGRVU 0QY KPXCTKCPV UGEQPFNGXGN EQPEGRVU COQWPV VQ YJCV KP CDUVTCEV OQFGN VJGQT[ CTG ECNNGF
IGPGTCNK\GF SWCPVKHKGTU
6JGUG KPENWFG HQT KPUVCPEG
PQPG UQOG CNN CV OQUV QPG
GZCEVN[ QPG CV NGCUV QPG CV OQUV VYQ GZCEVN[ VYQ CV NGCUV VYQ HKPKVGN[ OCP[ KPHKPKVGN[ OCP[ OQUV GXGPN[ OCP[ QFFN[ OCP[
NQIKECN
CPF UQ QP 6JG CUUWORVKQP QH VJG GZKUVGPEG QH
EQTTGURQPFKPI VQ VJGUG SWCPVKHKGTU CV QPEG IKXGU WU UWTTQICVGU HQT PCVWTCN
QDLGEVU
PWODGTU KP VJG HQTO QH VJG QDLGEVU EQTTGURQPFKPI VQ VYQ
GZCEVN[ \GTQ GZCEVN[ QPG GZCEVN[
CPF UQ QP 4GEQIPK\KPI VJGUG QDLGEVU EQTTGURQPFU VQ VJG JKUVQTKECN UVGR QH UYKVEJKPI
HTQO VJG CFLGEVKXCN WUG QH PWOGTCNU CU KP WUG QH PWOGTCNU CU KP
Q
Q
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R
R
VQ VJG PQOKPCN
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CPF
QFFN[ OCP[
ECP UGTXG CU UWTTQICVGU HQT VJG UGVU QH GXGP CPF QH QFF
O
PWODGTU 5WEJ KU VJG KFGC QH (KPG U KPVGTRTGVCVKQP QH CPCN[UKU YKVJ TGCN PWODGTU CU
QDLGEVU
KP JKU VJGQT[ C UQOGYJCV OQTG TKIQTQWU VTGCVOGPV YKNN DG IKXGP UJQTVN[ 6JGTG KU CNUQ C UGEQPFQTFGT XGTUKQP 9)6# +P VJKU VJGQT[ YG JCXG C TWNG RGTOKVVKPI WU HQT CP[ HQTOWNC YKVJ
θ : ; VQ KPVTQFWEG C PGY U[ODQN h CPF CUUWOG # #
θ KP RNCEG QH 4 6JG HQTOWNC θ OC[ EQPVCKP RCTCOGVGTU VJCV KU KV OC[ NQQM NKMG
θ 7 W : ; KP YJKEJ ECUG h YKNN JCXG GZVTC CTIWOGPV RNCEGU HQT VJGUG RCTCOGVGTU CPF
7W: HQT h:7W %QTTGURQPFKPI VQ GZVGPUKQPCNKV[ KP VJG VJKTFQTFGT ECUG YG
YG YKNN YTKVG h
CUUWOG HQT HQTOWNCU
θ CPF χ CPF CUUQEKCVGF U[ODQNU h CPF { VJG HQNNQYKPI
∀:∀; θ 7 W : ; ↔ χ 8 X : ; → h7W: {8X;
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θ θ θ
W
VJGP KPVTQFWEG h CUUQEKCVGF YKVJ
HQTOWNCU YG IGV YKVJ VJKU U[ODQN CU
θ θ θ
W
θ VJGP GPWOGTCVG CNN VJG PGY
VJGP KPVTQFWEG h CUUQEKCVGF YKVJ
θ
VJGP GPWOGTCVG VJG PGY HQTOWNCU YG IGV YKVJ VJKU U[ODQN CU
θ θ θ
W
CPF
EQPVKPWG KP VJKU YC[ GCEJ VKOG WUKPI WR JCNH VJG TGOCKPKPI RNCEGU 6JG TGUWNV KU C
K
NCPIWCIG YKVJ C U[ODQN h HQT GXGT[ HQTOWNC L
θK CPF YKVJ GCEJ θK EQPVCKPKPI QPN[ hL HQT
K # TKIQTQWU VTGCVOGPV QH VJG KPVGTRTGVCVKQP QH CPCN[UKU UJQYU VJCV 2 KU KPVGTRTGVCDNG
KP 9)6# WUKPI QPN[ HQTOWNC
VYQ
URGEKCN U[ODQNU 9G HKTUV PQVG VJCV HQT CP[
7
VJG HQNNQYKPI
φ 7 : ; FGVGTOKPGU CP GSWKXCNGPEG
:
≈ 7 ; ≈ 7 ∨ `: ≈ 7 `; ≈ 7
0QVG VJCV HQT CP[ RGTOWVCVKQP
2
CPF CP[
:
YG JCXG
2 :
≈ : 6JGP C NKVVNG VJQWIJV UJQYU
M
φ 7 : ; FGVGTOKPGU C FQWDN[ KPXCTKCPV GSWKXCNGPEG CPF VJG UCOG QPG HQT VYQ FKHHGTGPV RCTCOGVGTU 7 CPF 7 KH CPF QPN[ KH 7 ≈ 7 1DXKQWUN[ KV KU C FKEJQVQO[ 5Q
VJCV
φ 7 : ; KU UWKVCDNG CPF YG OC[ KPVTQFWEG C U[ODQN
T7 T7 :
;
T
CPF CUUWOG VJG HQNNQYKPI
↔ : ≈ 7 ; ≈ 7 ∨ `: ≈ 7 `; ≈ 7
# NKVVNG VJQWIJV UJQYU VJCV FGHKPKPI 7 9G PGZV PQVG VJCV HQT CP[
7
T7
7
O
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7 :
7 ;
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CPF CP[
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FKEJQVQO[ 5Q
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# NKVVNG VJQWIJV UJQYU VJCV KORNKGU VJCV HQT CP[
7
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n P UVQTFGT 2GCPQ CTKVJOGVKE ECP DG
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4 KU CP GSWKXCNGPEG
4 KU XGT[ PGCTN[ KPXCTKCPV
4 KU PQPKPHNCVKPI
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QXGT[
R KP RNCEG QH QUOCNNR 9KVJ VJKU YGCMGPGF FGHKPKVKQP QH UWKVCDKNKV[ YG IGV VJG
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EQWTUG GPQWIJ #PF UQ HQT RTGUGPV RWTRQUGU KV YKNN DG GPQWIJ VQ GZVTCEV CPF EQPFGPUG LWUV
O
GPQWIJ QWV QH (KPG U GZVGPFGF FKUEWUUKQP VQ IKXG WU QPG OQFGN 7PHQTVWPCVGN[ HTQO VJKU RQKPV QP KV YKNN DG PGEGUUCT[ VQ CUUWOG OQTG HCOKNKCTKV[ YKVJ CZKQOCVKE UGV VJGQT[ VJCP + JCXG DGGP CUUWOKPI UQ HCT VJQWIJ VJG KPHQTOCVKQP VJCV ECP DG QDVCKPGF HTQO VJG GCTNKGT RCTVU QH CP[ KPVTQFWEVQT[ VGZV DQQM UJQWNF DG OQTG VJCP GPQWIJ +P EQPUVTWEVKPI C OQFGN VQ RTQXG EQPUKUVGPE[ HQT VJG UVTQPI IGPGTCN VJGQT[ QH CDUVTCEVKQP + YKNN CNUQ CUUWOG VJG CZKQO QH EQPUVTWEVKDKNKV[ 8 . #NN VJCV QPG PGGFU VQ MPQY CDQWV VJKU CZKQO KU VJCV KV KORNKGU VJG EQPVKPWWO J[RQVJGUKU %* CPF KORNKGU VJCV C
EGTVCKP TGNCVKQP . FGHKPCDNG D[ C EGTVCKP HQTOWNC
λ Z [ QH VJG NCPIWCIG QH UGV VJGQT[
IKXGU C YGNNQTFGT QH CNN UGVU 6JKU HGCVWTG ECP DG WUGF YJGPGXGT YG PGGF VQ OCMG C EJQKEG QH C UGV YKVJ C EGTVCKP RTQRGTV[ CPF YCPV VQ DG GZRNKEKV CDQWV YJKEJ EJQKEG YG CTG OCMKPI YG VJGP EJQQUG VJG .NGCUV UGV YKVJ VJG RTQRGTV[ $WV VJKU HGCVWTG YKNN DG WUGF QPN[ CV VJG XGT[ GPF QH VJG RTQQH 6JG CUUWORVKQP QH 8 . KU JCTONGUU (QT )ÑFGN UJQYGF VJCV <( 8 . KU C EQPUGTXCVKXG GZVGPUKQP QH <( HQT CTKVJOGVKECN UVCVGOGPVU YJKEJ KU VQ UC[ VJCV CP[ EQPENWUKQP VJCV EQWNF DG UVCVGF KP VJG NCPIWCIG QH HKTUVQTFGT CTKVJOGVKE VJCV KU RTQXGF YKVJ VJG CKF QH VJKU J[RQVJGUKU EQWNF DG RTQXGF YKVJQWV KV CPF EQPUKUVGPE[ UVCVGOGPVU ECP DG EQFGF KP VJG NCPIWCIG QH HKTUVQTFGT CTKVJOGVKE CU VJG[ CTG KP VGZVDQQMU EQPPGEVKQP YKVJ VJG UGEQPF KPEQORNGVGPGUU VJGQTGO
+P VJG OQFGN VQ DG EQPUVTWEVGF VJG FQOCKP QH VJG QDLGEV XCTKCDNGU YKNN DG
ω VJG
HKTUV WPEQWPVCDNG QTFKPCN KFGPVKHKGF YKVJ VJG UGV QH CNN EQWPVCDNG QTFKPCNU 6JG OQFGN YKNN DG UVCPFCTF UQ VJG FQOCKP QH VJG QPGRNCEG EQPEGRV XCTKCDNGU YKNN DG RQYGT UGV
℘ ω QT
ω CPF VJG FQOCKP QH VJG VYQRNCEG TGNCVKQPCN EQPEGRV XCTKCDNGU YKNN DG VJG RQYGT UGV ℘ ω ⊗ ω QH VJG %CTVGUKCP RTQFWEV ω ⊗ ω QT UGV QH CNN QTFGTGF RCKTU UGV QH CNN UWDUGVU QH
QH GNGOGPVU QH
ω VJCV KU VJG UGV QH CNN VYQRNCEG TGNCVKQPU QP ω +P VJG VJKTFQTFGT
XGTUKQP QH VJG VJGQT[ VJG UGEQPFNGXGN QPGRNCEG EQPEGRV XCTKCDNGU CPF VYQRNCEG TGNCVKQPCN EQPEGRV XCTKCDNGU YKNN TCPIG QXGT UWDUGVU QH
℘ ω CPF QH ℘ ω ⊗ ℘ ω
5KPEG VJG OQFGN KU UVCPFCTF CNN EQORTGJGPUKQP CZKQOU QH JKIJGTQTFGT NQIKE YKNN JQNF CU YKNN GZVGPUKQPCNKV[ +V TGOCKPU VQ EQPUKFGT VJG CZKQOU QH CDQWV CDUVTCEVU 9JCV YG PGGF VQ FQ VQ IGV C OQFGN KU VQ CUUQEKCVG VQ CP[ UWKVCDNG GSWKXCNGPEG 4 CPF CP[ :
⊆ ω CP QTFKPCN h4: ω
KP UWEJ CU YC[ VJCV YG JCXG
↔ h4: h4;
4:;
h : h ;
4
5 → ∀< 4:< ↔ 5;<
6JG EQPUVTWEVKQP ECP CEVWCNN[ DG UQ CTTCPIGF CU VQ IKXG YJKEJGXGT QH VJG HQNNQYKPI KPEQORCVKDNG UVTGPIVJGPKPIU QH QPG RTGHGTU
C
D
l 5 → h4: l h5; h4: h5; ↔ ∀< 4:< ↔ 5;< 4
(QT VJG EQPUVTWEVKQP YG YKNN QH EQWTUG DG WUKPI VJG UWKVCDKNKV[ J[RQVJGUKU QP VJG GSWKXCNGPEG 4 .GV WU HKTUV EQPUKFGT YJCV
QXGT[ PGCTN[ KPXCTKCPVR COQWPVU VQ KP QWT
RTGUGPV EQPVGZV 5KPEG VJG FQOCKP QH VJG QDLGEV XCTKCDNGU JCU UK\G
ℵ QUOCNNR YKNN OGCP
ℵ YJKEJ KU VQ UC[ m ℵ QT EQWPVCDNG CPF QXGT[ UOCNNR YKNN OGCP ℵ QT HKPKVG 5Q
KH 4 KU XGT[ PGCTN[ KPXCTKCPV VJGTG YKNN DG C HKPKVG UWDUGV U QH
ω UWEJ VJCV 4 KU UKPXCTKCPV
QPQPKPHNCVKPIR YCU C VGEJPKECN EQPFKVKQP UWHHKEKGPV CPF CEVWCNN[ PGEGUUCT[ VJQWIJ YG YKNN PQV PGGF VJCV HCEV VQ KORN[ JCXKPI m ℵ UGVU QH GSWKXCNGPVU 6JG FGHKPKVKQP QH
9JCV KU ETWEKCN VQ VJG EQPUVTWEVKQP VQ GUVCDNKUJ VJCV VJGTG CTG QPN[
ℵ RCKTU
EQPUKUVKPI QH C UWKVCDNG GSWKXCNGPEG 4 CPF QPG QH KVU UGVU QH GSWKXCNGPVU 4=:? 6JG PQPKPHNCVKPI EQPFKVKQP VGNNU WU VJGTG YKNN PQV DG VQQ OCP[ UGVU QH GSWKXCNGPVU HQT CP[ IKXGP UWKVCDNG GSWKXCNGPEG UQ YJCV KU ETWEKCN YKNN DG VQ GUVCDNKUJ VJCV VJGTG CTG PQV VQQ OCP[ XGT[ PGCTN[ KPXCTKCPV GSWKXCNGPEGU 6JG MG[ VQ FQKPI UQ YKNN DG NQECVKPI C EQPFKVKQP
⊆ ω CPF : ; : ; ⊆ ω YKNN DG UWHHKEKGPV VQ KORN[ VJCV VJGTG GZKUVU C RGTOWVCVKQP 2 QH ω VJCV HKZGU U CPF JCU 2 : : CPF 2 ; ; VJCV HQT CP[ HKPKVG U
6JG MG[ VQ NQECVKPI UWEJ C EQPFKVKQP YKNN DG VJG HQNNQYKPI VGEJPKECN PQVKQP %CNN R C RTQHKNG KH R KU CP QEVWRNG YJQUG HKTUV HQWT EQORQPGPVU CTG HKPKVG UWDUGVU QH
ω CPF YJQUG
m ℵ CPF ECNN R CP URTQHKNG KH VJG WPKQP QH KVU HKTUV HQWT EQORQPGPVU KU U (QT IKXGP : CPF ; FGHKPG VJGKT URTQHKNG KU : ; VQ DG VJG QPG YKVJ
NCUV HQWT EQORQPGPVU CTG ECTFKPCNU
VJG HQNNQYKPI GKIJV EQORQPGPVU
:
∩ ; ∩ U
^^ :
∩ ;
: ;
U^^
∩
; :
U
^^ : ; U^^
6JGP + ENCKO VJCV KH
∩U
^^ ; : U^^
ω : ;
∩
U
^^ ω : ; U^^
KU : ; KU : ; VJGP VJGTG KU C RGTOWVCVKQP 2 QH ω VJCV HKZGU
U CPF JCU 2 : : CPF 2 ; ; 6Q RTQXG VJKU ENCKO DGIKP D[ PQVKPI VJCV VJG UCOGPGUU QH VJG HKTUV HQWT EQORQPGPVU QH VJG URTQHKNGU KORNKGU VJCV VJG KFGPVKV[ +U UCVKUHKGU +U :
∩ U : ∩
U CPF +U ;
GPQWIJ VQ HKPF C RGTOWVCVKQP 3 QH
∩ U ; ∩ U
6Q QDVCKP VJG FGUKTGF 2 KV YKNN DG
ω U YKVJ 3 : U : U CPF 3 ; U ; U HQT
VJGP +U CPF 3 ECP DG EQODKPGF KP VJG QDXKQWU YC[ VQ IKXG 2 #PF VJG FGUKTGF 3 ECP DG QDVCKPGF D[ EQODKPKPI HQWT QPGVQQPG EQTTGURQPFGPEGU QPG DGVYGGP :
:
∩ ;
∩ ;
U CPF
U CPQVJGT DGVYGGP : ; U CPF : ; U CPQVJGT DGVYGGP ; : U
CPF ; : U CPF CPQVJGT DGVYGGP ω : ; U CPF ω : ; U 6JGUG HQWT GZKUV D[ VJG UCOGPGUU QH VJG NCUV HQWT EQORQPGPVU QH VJG URTQHKNGU 0QY ECNN C UGV 5 HCKT KH HQT UQOG HKPKVG U
⊆ ω 5 KU C PQPGORV[ UGV QH URTQHKNGU
6JG U KP SWGUVKQP YKNN DG WPKSWG CPF ECP DG TGEQXGTGF HTQO 5 CU VJG WPKQP QH VJG HKTUV HQWT
K5 : ; VQ DG UKORN[ KU : ; HQT VJKU U #PF NGV WU FGHKPG : CPF ; VQ DG 5TGNCVGF KH K5 : ; ∈ 5 + ENCKO VJCV
EQORQPGPVU QH CP[ RTQHKNG R KP 5 .GV WU FGHKPG VJG 5RTQHKNG
GXGT[ XGT[ PGCTN[ KPXCTKCPV GSWKXCNGPEG 4 EQKPEKFGU YKVJ 5TGNCVGFPGUU HQT UQOG HCKT 5
6Q RTQXG VJG ENCKO NGV 4 DG IKXGP 6JGP UKPEG 4 KU XGT[ PGCTN[ KPXCTKCPV VJGTG KU C HKPKVG U
⊆ ω UWEJ VJCV 4 KU UKPXCTKCPV OGCPKPI VJCV HQT CP[ RGTOWVCVKQP 2 HKZKPI U YG JCXG
4:; KH CPF QPN[ KH 42 : 2 ; HQT CNN : CPF ; .GV 5 DG VJG UGV QH URTQHKNGU
KU : ; HQT :
CPF ; UWEJ VJCV 4:; 4GXKGYKPI VJG FGHKPKVKQPU KV KU KOOGFKCVG VJCV 4:; KORNKGU VJCV : CPF ; CTG 5TGNCVGF 6Q EQORNGVG VJG RTQQH QH VJG ENCKO KV YKNN DG GPQWIJ VQ UJQY VJCV VJG EQPXGTUG JQNFU 5Q UWRRQUG : CPF ; CTG 5TGNCVGF OGCPKPI VJCV CPF VJGTGHQTG D[ VJG FGHKPKVKQP QH 5 KORN[KPI VJCV ; UWEJ VJCV 4:; $WV UKPEG
KU : ; K5 : ; ∈ 5
KU : ; KU : ; HQT UQOG : CPF
KU : ; KU : ; VJGTG GZKUVU C RGTOWVCVKQP 2 VJCV HKZGU
U CPF JCU 2 : : CPF 2 ; ; HTQO YJKEJ KV HQNNQYU D[ VJG UKPXCTKCPEG QH 4 VJCV YG JCXG 4:; CU TGSWKTGF VQ EQORNGVG VJG RTQQH QH VJG ENCKO %CNN C UGV 5 IQQF KH 5 KU HCKT CPF 5TGNCVGFPGUU KU C UWKVCDNG GSWKXCNGPEG 6JGP YJCV JCU LWUV DGGP UJQYP KU VJCV HQT GXGT[ UWKVCDNG GSWKXCNGPEG 4 VJGTG KU C IQQF 5 UWEJ VJCV 4 EQKPEKFGU YKVJ 5TGNCVGFPGUU #PF YJCV YG JCXG CEEQORNKUJGF UQ HCT KU VQ UJQY VJCV VJGTG CTG PQ OQTG UWKVCDNG GSWKXCNGPEGU 4 VJCP VJGTG CTG IQQF 5 9G PGGF PGZV C DQWPF QP VJG PWODGT QH UWEJ 5 + ENCKO VJGTG CTG QPN[
ℵ HCKT 5 CPF C HQTVKQTK QPN[ ℵ IQQF 5
5KPEG GCEJ HCKT 5 KU C UGV QH URTQHKNGU HQT UQOG HKPKVG U
⊆ ω CPF VJGTG CTG QPN[ ℵ
UWEJ U VQ RTQXG VJG ENCKO KV YKNN DG GPQWIJ VQ UJQY VJCV HQT CP[ IKXGP U VJG PWODGT QH UGVU 5 QH URTQHKNGU KU PQ OQTG VJCP
ℵ 0QY GXGT[ UWEJ 5 KU C UWDUGV QH VJG UGV QH CNN
URTQHKNGU R CPF VJG UGV QH CNN URTQHKNGU KU EQWPVCDNG (QT GCEJ QH VJG HKTUV HQWT EQORQPGPVU QH UWEJ C R KU C UWDUGV QH U CPF VJGTG CTG QPN[ HKPKVGN[ OCP[ QH VJQUG YJKNG GCEJ QH VJG NCUV HQWT EQORQPGPVU QH UWEJ C R KU C ECTFKPCN
m ℵ CPF VJGTG CTG QPN[
EQWPVCDN[ OCP[ QH VJQUG PCOGN[ VJG PCVWTCN PWODGTU CPF
ℵ CPF ℵ 6Q EQORNGVG VJG
RTQQH QH VJG DQWPF QP VJG PWODGT QH XGT[ IQQF 5 YG KPXQMG HQT VJG HKTUV CPF NCUV VKOG %* YJKEJ VGNNU WU VJCV VJG PWODGT QH UWDUGVU QH C EQWPVCDNG UGV KU
ℵ
6Q EQORNGVG VJG RTQQH QH EQPUKUVGPE[ KV YQWNF DG GPQWIJ PQY VQ CUUKIP GCEJ RCKT
〈4 :〉 EQPUKUVKPI QH C UWKVCDNG GSWKXCNGPEG 4 CPF QPG QH KVU UGVU QH GSWKXCNGPVU : C FKUVKPEV QTFKPCN ω YJKEJ YG PQY MPQY KU RQUUKDNG UKPEG YG JCXG UGGP VJCV VJGTG CTG ℵ UWKVCDNG GSWKXCNGPEGU CPF VJGTG CTG CV OQUV ℵ UGVU QH GSWKXCNGPVU HQT GCEJ OCMKPI QPN[ ℵ UWEJ RCKTU KP CNN 6JKU FQPG YG EQWNF VCMG h4: VQ VJG QTFKPCN CUUKIPGF VQ
QPN[
〈4 4=:?〉 CPF YG YQWNF JCXG C OQFGN QH CPF C # UNKIJVN[ OQFKHKGF EQPUVTWEVKQP CUUKIPKPI QTFKPCNU QPN[ VQ VJG UGVU QH GSWKXCNGPVU : TCVJGT VJCP RCKTU 〈 4 :〉 YQWNF IKXG
D KPUVGCF 9G YQWNF JCXG EQPUKUVGPE[ HQT GKVJGT QRVKQP CPF YKVJQWV WUKPI OQTG QH 8 . VJCP KVU KORNKECVKQP %*
6JQWIJ KPFGGF VJKU KU GPQWIJ VQ RTQXG EQPUKUVGPE[ KV YKNN DG YQTVJYJKNG VQ OCMG
4
WUG QH . CPF IKXG C OQTG GZRNKEKV FGHKPKVKQP QH h : 6Q DGIKP YKVJ ECNN 5 XGT[ IQQF KH
5 KU IQQF CPF VJGTG KU PQ 6 . 5 UWEJ VJCV 6 KU IQQF CPF 6TGNCVGFPGUU EQKPEKFGU YKVJ 5TGNCVGFPGUU 6JGP GXGT[ UWKVCDNG 4 EQKPEKFGU YKVJ 5TGNCVGFPGUU HQT UQOG WPKSWG XGT[ IQQF 5 %CNN C RCKT 〈 5 ;〉 GZEGNNGPV KH 5 KU XGT[ IQQF ;
⊆ ω CPF ; KU VJG .NGCUV OGODGT
QH KVU UGV QH GSWKXCNGPVU WPFGT 5TGNCVGFPGUU QT KP QVJGT YQTFU VJGTG KU PQ < . ; UWEJ VJCV ; CPF < CTG 5TGNCVGF 5KPEG 5TGNCVGFPGUU KU C PQPKPHNCVKPI GSWKXCNGPEG VJGTG CTG CV OQUV VJGTG
ℵ UGVU QH GSWKXCNGPVU GCEJ QH YJKEJ JCU GZCEVN[ QPG .NGCUV OGODGT UQ VJCV KP CNN
m ℵ GZEGNNGPV RCKTU 〈5 ;〉 HQT CP[ QPG QH VJG ℵ XGT[ IQQF 5 CPF JGPEG QPN[ ℵ
GZEGNNGPV RCKTU CNVQIGVJGT /QTGQXGT HQT CP[ RCKT 〈 4 :〉 EQPUKUVKPI QH C UWKVCDNG GSWKXCNGPEG 4 CPF C UWDUGV QH
ω VJGTG KU C WPKSWG GZEGNNGPV RCKT 〈5 ;〉 UWEJ VJCV
4GSWKXCNGPEG EQKPEKFGU YKVJ 5TGNCVGFPGUU CPF : KU 5TGNCVGF VQ ; YJKEJ KU VQ UC[ : KU 4GSWKXCNGPV VQ ; 9G CNTGCF[ MPQY VJGTG KU C WPKSWG XGT[ IQQF 5 UWEJ VJCV 4GSWKXCNGPEG EQKPEKFGU YKVJ 5TGNCVGFPGUU CPF KV KU CNUQ ENGCTN[ VJG ECUG VJCV VJGTG KU C WPKSWG ; VJCV KU .NGCUV COQPI CNN UWDUGVU QH
ω VJCV CTG 4GSWKXCNGPV VQ : .GV WU UC[
VJCV UWEJ C RCKT ECRVWTGU 4 CPF : 5KPEG VJG PWODGT QH GZEGNNGPV RCKTU KU QPGVQQPG EQTTGURQPFGPEG DGVYGGP QTFKPCNU
ℵ VJGTG KU C
ω CPF GZEGNNGPV RCKTU CPF COQPI CNN UWEJ
EQTTGURQPFGPEGU VJGTG KU QPG VJCV KU .NGCUV YJKEJ YG OC[ ECNN VJG ECPQPKECN EQTTGURQPFGPEG 6JGP HQT CP[ RCKT 〈 4 :〉 EQPUKUVKPI QH C UWKVCDNG GSWKXCNGPEG 4 CPF C UWDUGV QH
ω VJGTG KU C WPKSWG QTFKPCN Z ω UWEJ VJCV Z EQTTGURQPFU WPFGT VJG ECPQPKECN
EQTTGURQPFGPEG VQ VJG GZEGNNGPV RCKT VJCV ECRVWTGU 4 CPF : .GV
σ 4 : DG VJKU WPKSWG
QTFKPCN 4GXKGYKPI VJG FGHKPKVKQPU YG UGG VJCV KH 4 CPF 5 FQ PQV EQKPEKFG VJGP
σ 4 : l σ 5 ; HQT CP[ : CPF ; YJKNG HQT HKZGF 4 YG JCXG σ 4 : σ 4 ; KH CPF QPN[ KH : CPF ; CTG 4GSWKXCNGPV 6JGTGHQTG KH YG VCMG h4: VQ FGPQVG σ 4 ; YG IGV C OQFGN QH CPF C # UNKIJVN[ OQFKHKGF EQPUVTWEVKQP YQWNF IKXG D KPUVGCF 6JKU TGCNN[ KU GPQWIJ VQ EQORNGVG VJG RTQQH QH EQPUKUVGPE[ DWV NQQMGF CV ENQUGN[ VJG RTQQH CEVWCNN[ RTQXGU OQTG VJCP EQPUKUVGPE[ 6JG FGHKPKVKQP QH C OQFGN KP GHHGEV IKXGU CP KPVGTRTGVCVKQP QH 5)6# KP <(% 8 . /QTGQXGT VJQWIJ VJG RTQQH WUGF VJG GZKUVGPEG QH
ω CPF ℘ ω VQ IGV FQOCKPU HQT VJG QDLGEV CPF HKTUVNGXGN EQPEGRVXCTKCDNGU
CPF VJGTGD[ WUGF VJG GZKUVGPEG QH
℘ ω CPF ℘ ω CPF VJQWIJ VJG RTQQH WUGF UWDUGVU QH
℘ ω CU VJG FQOCKP QH VJG UGEQPFNGXGN EQPEGRV XCTKCDNGU VJG KPVGTRTGVCVKQP PQYJGTG TGCNN[ WUGU VJG CUUWORVKQP VJCV VJGUG UWDUGVU HQTO C UGV YJKEJ KU VQ UC[ VJCV YG FQ PQV PGGF
℘ ω QT ℘ ω HQT VJG RTQQH 6JKU OGCPU VJCV YG CEVWCNN[ IGV CP KPVGTRTGVCVKQP KP
<(
℘ ω 8 . #PF KP QWT RTGNKOKPCT[ UWTXG[ QH U[UVGOU QH CPCN[UKU CPF UGV
VJGQT[ KV YCU TGOCTMGF VJCV VJKU VJGQT[ KU KPVGTRTGVCDNG KP <(
℘ ω YJKEJ KP VWTP KU
KPVGTRTGVCDNG KP 2 6JKU IKXGU VJG ECUG P KP VJG HQNNQYKPI VJGQTGO HTQO YJKEJ VJG ECUGU HQT P
HQNNQY VTKXKCNN[
%QPXGTUG VQ VJG (TGIG(KPG 6JGQTGO (QT CP[ P
n VJG PVJQTFGT UVTQPI IGPGTCN
VJGQT[ QH CDUVTCEVU KU KPVGTRTGVCDNG KP P UVQTFGT 2GCPQ CTKVJOGVKE
6JG ECUG P KU C NKVVNG OQTG FGNKECVG CPF + YKNN PQV FQV GXGT[ K QT ETQUU GXGT[ V (QT VJG EQIPQUEGPVK YJCV VJG EQPUVTWEVKQP KP KVU OQTG GZRNKEKV XGTUKQP WUKPI . TGCNN[ UJQYU KU VJCV VJGTG KU C YC[ VQ CUUQEKCVG VQ CP[ HQTOWNC
θ 7 W : ; QH VJG NCPIWCIG QH
θ 7 W : Z QH VJG NCPIWCIG QH UGV VJGQT[ UWEJ VJCV HQT CP[ 7 CPF W KH θ 7 W : ; FGHKPGU C UWKVCDNG GSWKXCNGPEG QP UWDUGVU QH ω VJGP θ 7 W : Z
UGV VJGQT[ C HQTOWNC
ω VQ UWDUGVU : ⊆ ω UWEJ VJCV CPF C JQNF YKVJ θ 7 W : ; HQT 4:; CPF VJG WPKSWG Z UWEJ VJCV θ 7 W : Z CU h4: 6JG HQTOWNC FGHKPGU CP CUUKIPOGPV QH CDUVTCEVU Z
θ UC[U VJCV NGVVKPI σ DG VJG ECPQPKECN GPWOGTCVKQP QH GZEGNNGPV RCKTU CPF 〈5 ;〉 σ Z VJGP VJG RCKT 〈 5 ;〉 ECRVWTGU VJG RCKT EQPUKUVKPI QH VJG TGNCVKQP FGHKPGF D[ θ 7 W : ; VQIGVJGT YKVJ : 6JCV VJG TGNCVKQP FGHKPGF D[ TGSWKTGF #PF VJWU KP <(
θ 7 W : ; GZKUVU CU C UGV 4 KU PQV TGCNN[
℘ ω CPF ℘ ω CTG PQV TGCNN[ TGSWKTGF CPF YG IGV CP KPVGTRTGVCVKQP
℘ ω 8 . CPF VJGPEG KP <( ℘ ω CPF VJGPEG KP 2
6JG HKPCN TGUWNV KU VJCV KH YG UVC[ YKVJKP VJG DQWPFU QH UGEQPFQTFGT NQIKE CU OQUV PGQ(TGIGCPU FQ VJGP TGN[KPI QP KPXCTKCPEG VQ UQNXG VJG J[RGTKPHNCVKQP RTQDNGO VJG
QNKOKVU QH CDUVTCEVKQPR VWTP QWV VQ DG VJQUG QH VJKTFQTFGT 2GCPQ CTKVJOGVKE 6JKU KU TGCNN[ C SWKVG UVTQPI VJGQT[ CPF CP[ OCVJGOCVKEU VJCV ECPPQV DG FQPG KP KV OKIJV YGNN DG ECNNGF
QJKIJGT UGV VJGQT[R UQ VJCV (KPGOU CRRTQCEJ ECP DG UCKF VQ IGV WU CNN QH OCVJGOCVKEU GZEGRV JKIJGT UGV VJGQT[
O
7PHQTVWPCVGN[ JQYGXGT (KPG U CRRTQCEJ FQGU PQV IKXG C EQORNGVG CPUYGT VQ VJG DCFEQORCP[ QDLGEVKQP CU (KPG JKOUGNH KP GHHGEV EQPEGFGU 6JG RTQDNGO KU VJCV HQT CP[ 4 VJG TGNCVKQP FGHKPGF CU HQNNQYU
'456
↔ 5 ≅ 4 6 ≅ 4 ∨ `5 ≅ 4 `6 ≅ 4
KU KPXCTKCPV CPF C FKEJQVQO[ CPF [GV VQ CUUWOG CDUVTCEVU GZKUV HQT CNN UWEJ GSWKXCNGPEGU KU VQ CUUWOG VJG GZKUVGPEG QH KUQOQTRJKUO V[RGU KPENWFKPI QTFGT V[RGU KPENWFKPI QTFKPCN
O
PWODGTU CPF VJG $WTCNK(QTVK RCTCFQZ GPUWGU #P CRRTQRTKCVG GZVGPUKQP QH (KPG U CRRTQCEJ HTQO GSWKXCNGPEGU QP QPGRNCEG EQPEGRVU VQ GSWKXCNGPEGU QP VYQRNCEG TGNCVKQPCN EQPEGRVU JCU PQV [GV DGGP FGXGNQRGF
0GQ(TGIGCP5GV6JGQTKGU 6JG EQOOQP HGCVWTG QH VJG 9TKIJV*CNG CPF (KPG CDUVTCEVKQPKUV CRRTQCEJGU KU VJCV VJG[ FKURGPUG YKVJ .CY 8 CPF GZVGPUKQPU CPF YQTM YKVJ CDUVTCEVU HQT GSWKXCNGPEGU VJCV CTG SWKVG FKHHGTGPV HTQO EQGZVGPUKXGPGUU + YKUJ PQY VQ VWTP VQ VJGQTKGU VJCV KPUVGCF UCXG (TGIG HTQO EQPVTCFKEVKQP D[ TGUVTKEVKPI .CY 8 5WEJ VJGQTKGU MGGR GZVGPUKQPU VJQWIJ
O
UWDLGEV VQ C TGUVTKEVKQP CPF NKMG (TGIG U QTKIKPCN VJGQT[ VJG[ FQ PQV FKTGEVN[ CUUWOG CDUVTCEVU HQT CP[ GSWKXCNGPEG QVJGT VJCP EQGZVGPUKXGPGUU +P C UGPUG QH EQWTUG CP[ CZKQOCVKE UGV VJGQT[ ECP DG XKGYGF CU C TGUVTKEVKQP QP
O
(TGIG U KPEQPUKUVGPV UGV VJGQT[ 6JG UGV VJGQTKGU + EQPUKFGT PGQ(TGIGCP JQYGXGT JCXG
O
VYQ HGCVWTGU VJCV OCMG VJGO UKOKNCT VQ (TGIG U QTKIKPCN VJGQT[ CPF FKHHGTGPV HTQO OCKPUVTGCO CZKQOCVKE UGV VJGQTKGU UWEJ CU <(% 6JG HKTUV UWEJ HGCVWTG KU VJCV VJG[ OCMG
O
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e CPF ² CPF
∈
YKVJ VJG HQNNQYKPI CZKQOU (KTUV YG JCXG VJG CZKQOU QH UWDQTFKPCVKQP FGHKPKPI UGV CPF GNGOGPV KP VGTOU QH HCNNKPIWPFGT CPF GZVGPUKQPUQH
²Z
Z
↔ ∃: e Z:
∈ [ ↔ ∃; e [; ;Z
5GEQPF YG JCXG VJG CZKQO QH GZVGPUKQPCNKV[ KP VJG HQNNQYKPI (TGIGCP HQTO
e Z: e [;
→ Z [ ↔ : ≡ ;
0QVG VJCV KORNKGU VJCV VJG : KP CPF VJG ; KP CTG WPKSWG 9G JCXG UGGP JQY KORNKGU VJG WPKSWGPGUU QH VJG GZVGPUKQP YJGP KV GZKUVU RGTOKVVKPI VJG FGHKPKVKQP QH ]Z
U: CPF
φ Z _ HQT VJQUG EQPEGRVU : CPF nZ φ Z VJCV JCXG GZVGPUKQPU #PF YG JCXG UGGP JQY
RGTOKV VJG FGFWEVKQP COQPI QVJGT VJKPIU QH VJG HQNNQYKPI CNVGTPCVG HQTOWNCVKQP QH GZVGPUKQPCNKV[
²Z ²[
∀\ \ ∈ Z ↔ \ ∈ [ → Z [
6JGP YG JCXG C UGVGZKUVGPEG CZKQO QH QPG QH VJG VYQ HQTOU
C
∃Z e Z: ↔ : UCHG
D
: UCHG
→ ∃Z e Z:
+ YKNN EQPEGPVTCVG QP VJQUG YKVJ CZKQOU KP DKEQPFKVKQPCN HQTO C 9G JCXG UGGP JQY UWEJ C TGUVTKEVGF GZKUVGPEG CUUWORVKQP EQWNF DG CTVKHKEKCNN[ RTGUGPVGF CU VJG CUUWORVKQP QH VJG GZKUVGPEG QH CDUVTCEVU HQT C EGTVCKP GSWKXCNGPEG CITGGKPI YKVJ
≡ HQT UCHG : DWV
EQWPVKPI CNN WPUCHG : CU GSWKXCNGPV CPF VJCV CFQRVKPI VJKU CTVKHKEKCN RTGUGPVCVKQP KU GSWKXCNGPV VQ CUUWOKPI QPG CFFKVKQPCN PCOGCDNG QDLGEV
ο PQV VJG GZVGPUKQP QH CP[
EQPEGRV
1PG V[RG QH UGV VJGQT[ HKVVKPI VJKU HQTOCV YG JCXG EQPUKFGTGF CNTGCF[ VJG \KI\CI VJGQTKGU YJGTG UCHGV[ KU WPFGTUVQQF CU C MKPF QH UKORNKEKV[ +P VJKU UGEVKQP YG YKNN JCXG C NQQM CV QVJGT V[RGU 1PG UWEJ JCU DGGP UWIIGUVGF D[ (KPG 6JG UWIIGUVKQP KU VJCV YG VCMG
QUCHGR VQ OGCP QPGCTN[ KPXCTKCPVR UQ VJG GZKUVGPEG CZKQO TGCFU CU HQNNQYU
∃Z e Z: ↔ : PGCTN[ KPXCTKCPV # NKVVNG VJQWIJV UJQYU VJCV KH : KU PGCTN[ KPXCTKCPV UQ VJCV VJGTG KU UQOG UOCNN 9
UWEJ VJCV : KU 9KPXCTKCPV VJGP GKVJGT CNN Z QWVUKFG 9 JCXG `:Z QT CNN Z QWVUKFG 9 JCXG :Z 6JKU OGCPU VJCV GKVJGT : KVUGNH KU UOCNN QT KVU EQORNGOGPV KU UOCNN KP YJKEJ ECUG YG UC[ : KU EQUOCNN +V GCUKN[ UGGP VJCV EQPXGTUGN[ CP[ UGV VJCV KU UOCNN QT EQUOCNN KU PGCTN[ KPXCTKCPV 6JWU VJG UGV GZKUVGPEG CZKQO KU GSWKXCNGPV VQ VJG HQNNQYKPI
∃Z e Z: ↔ : UOCNN ∨ : EQUOCNN 6JG (TGIGCP UGV VJGQT[ YKVJ VJKU CZKQO + YKNN ECNN (KPG UGV VJGQT[ +V KU
KPVGTRTGVCDNG KP UGEQPFQTFGT 2GCPQ CTKVJOGVKE (QT KP 2 UOCNN COQWPVU VQ HKPKVG CPF VJGTG CTG UKORNG EQFKPIU QH HKPKVG UGVU QH PCVWTCN PWODGTU VJCV CTG FGHKPCDNG KP 2 CPF KPFGGF CNTGCF[ KP 2 CPF KPFGGF CNTGCF[ KP UVKNN YGCMGT VJGQTKGU CPF VJGUG ECP GCUKN[ DG GZVGPFGF VQ IKXG C EQFKPI QH HKPKVG CPF EQHKPKVG UGVU 1PG EQFKPI QH HKPKVG UGVU EQFGU U D[ VJG UWO
σ U QH VJG RQYGTU O QH HQT O ∈ U CPF VJKU ECP DG OQFKHKGF VQ C EQFKPI KP
YJKEJ
⋅ σ U EQFGU U CPF ⋅ σ U EQFGU VJG EQORNGOGPV QH U #PF VJKU KU CNN VJG
OCEJKPGT[ PGGFGF VQ IKXG CP KPVGTRTGVCVKQP %QPXGTUGN[ YQTMKPI KP (KPG UGV VJGQT[ UKPEG VJGTG KU CV NGCUV QPG QDLGEV CP GORV[ EQPEGRV KU UOCNN CPF JGPEG VJG GORV[ UGV
∅ GZKUVU CPF VJG WPKXGTUCN EQPEGRV KU EQUOCNN
CPF JGPEG VJG WPKXGTUCN UGV 8 GZKUVU /QTGQXGT UKPEG VJGTG KU CV NGCUV QPG QDLGEV 8
l ∅
CPF VJGTG CTG KP HCEV CV OQUV VYQ QDLGEVU 6JKU KU GPQWIJ VQ IKXG WU UKPINGVQPU CPF EQUKPINGVQPU CPF KP RCTVKEWNCT VJG CZKQOU QH 7756 YJKEJ YG MPQY VQ DG GPQWIJ YKVJ F[CFKE UGEQPFQTFGT NQIKE VQ RGTOKV VJG KPVGTRTGVCVKQP QH 2 + VCMG KV CNN VJKU YCU MPQYP VQ (KPG VJQWIJ PQV SWKVG GZRNKEKVN[ UVCVGF KP JKU DQQM
O
.GV OG UGV KV FQYP CU CP QHHKEKCN VJGQTGO YKVJ C PCOG VJCV YKNN FKUVKPIWKUJ KV HTQO (KPG U NCTIGT VJGQTGOU QP VJG KPVGTRTGVCDKNKV[ QH CPCN[UKU KP VJG IGPGTCN VJGQT[ QH CDUVTCEVKQP CPF VJG EQPUKUVGPE[ QH VJCV VJGQT[
(KPGOU .KVVNG 6JGQTGO (KPG UGV VJGQT[ CPF UGEQPFQTFGT 2GCPQ CTKVJOGVKE CTG KPVGTRTGVCDNG KP GCEJ QVJGT
6JG TGOCKPKPI V[RG QH UGV VJGQT[ VQ DG EQPUKFGTGF KU DCUGF QP CP KFGC CNTGCF[ OGPVKQPGF VJCV QH NKOKVCVKQP QH UK\G YJQUG TQNG KP VJG JKUVQT[ QH UGV VJGQT[ HTQO %CPVQT VQ VJG RTGUGPV FC[ JCU DGGP GZCOKPGF KP FGVCKN D[ /KEJCGN *CNNGVV + YKNN FKUVKPIWKUJ HQWT HQTOU QH VJG KFGC VYQ SWCPVKVCVKXG CPF VYQ PQPSWCPVKVCVKXG CPF VYQ OCZKOCNKUV CPF VYQ PQPOCZKOCNKUV +P CNN HQTOU VJG KFGC KU VJCV : JCU CP GZVGPUKQP KH CPF QPN[ KH VJGTG CTG PQV VQQ OCP[ QDLGEVU HCNNKPI WPFGT : 6JG SWCPVKVCVKXG XGTUKQPU CTG EQOOKVVGF VQ WPFGTUVCPFKPI
QOCP[R CPF QHGYR KP VGTOU QH QPGVQQPG EQTTGURQPFGPEGU
YJKNG VJG PQPSWCPVKVCVKXG CTG PQV 6JG OCZKOCNKUV XGTUKQPU CTG EQOOKVVGF VQ GSWCVKPI
QVQQ OCP[R YKVJ QCU OCP[ CU VJGTG CTG QDLGEVU CNVQIGVJGTR VJG PQPOCZKOCNKUV XGTUKQPU CTG PQV 6JG YGCMGUV QH VJG HQWT VJG PQPSWCPVKVCVKXG PQPOCZKOCNKUV XGTUKQP KU CNTGCF[
O
UWHHKEKGPV VQ OQVKXCVG
:
⊆ ; → ∃[ e [; → ∃Z e Z:
+H : KU C RCTV QH ; CPF VJGTG CTG PQV VQQ OCP[ QDLGEVU HCNNKPI WPFGT ; VJGP VJGTG CTG PQV VQQ OCP[ QDLGEVU HCNNKPI WPFGT : 5KPEG GXGT[ : KU EQPVCKPGF KP VJG WPKXGTUCN EQPEGRV KU KPEQORCVKDNG YKVJ VJG CUUWORVKQP VJCV VJG WPKXGTUCN EQPEGRV JCU CP GZVGPUKQP YJKEJ KU VQ UC[ YKVJ VJG CUUWORVKQP VJCV VJGTG GZKUVU C WPKXGTUCN UGV # NKVVNG VJQWIJV UJQYU VJCV KU
GSWKXCNGPV VQ VJG OQTG EQPXGPVKQPCN UGEQPFQTFGT UKPINGCZKQO HQTOWNCVKQP YJKEJ YKVJ EQORTGJGPUKQP KORNKGU GCEJ KPUVCPEG QH VJG CZKQOUEJGOG XGTUKQP
→ ∃Z ²Z ∀\ \ ∈ Z ↔ \ ∈ [ :\ ²[ → ∃Z ²Z ∀\ \ ∈ Z ↔ \ ∈ [ φ \
C
²[
D
O
6JG SWCPVKVCVKXG PQPOCZKOCNKUV XGTUKQP KU UWHHKEKGPV VQ OQVKXCVG (TCPMGN U CZKQO QH TGRNCEGOGPV YJKEJ ECP DG UVCVGF WUKPI F[CFKE UGEQPFQTFGT NQIKE VJWU
∀[∃Z4[Z ∀Z :Z ↔ ∃[ ;[ 4[Z → ∃[ e [; → ∃Z e Z:
#ICKP C NKVVNG VJQWIJV UJQYU VJCV KU GSWKXCNGPV VQ C OQTG EQPXGPVKQPCN UGEQPFQTFGT UKPINGCZKQO HQTOWNCVKQP YJKEJ KP VWTP KORNKGU GCEJ KPUVCPEG QH CP CZKQOUEJGOG XGTUKQP 6JG FGVCKNU YKNN PQV OCVVGT JGTG 6JG PQPSWCPVKVCVKXG OCZKOCNKUV XGTUKQP QH VJG NKOKVCVKQP QH UK\G KFGC YKNN DG QH PQ KOOGFKCVG EQPEGTP 6JG SWCPVKVCVKXG OCZKOCNKUV XGTUKQP OQVKXCVGU XQP 0GWOCPPOU CZKQO YJKEJ KP VJG RTGUGPV EQPVGZV OC[ DG HQTOWNCVGF CU HQNNQYU
∃Z e Z: ↔ : UOCNN
QUOCNNR KU FGHKPGF KP F[CFKE UGEQPFQTFGT NQIKE CU KV YCU GCTNKGT VQ OGCP QPQV KP QPGVQQPG EQTTGURQPFGPEG YKVJ VJG WPKXGTUG QH CNN QDLGEVUR $QQNQU EQPUKFGTGF VJG
*GTG
(TGIGCP UGV VJGQT[ YKVJ CU KVU UGVGZKUVGPEG CZKQO QT TCVJGT VJG CDUVTCEVKQPKUV XCTKCPV YJGTG YG JCXG QPG CFFKVKQPCN QDLGEV
ο FKUVKPEV HTQO CNN GZVGPUKQPU VJQWIJ JG WUGF VJKU
NCUV QPN[ VQ IGV VJG GZKUVGPEG QH CV NGCUV VYQ QDLGEVU .GV OG ECNN VJG UGV VJGQT[ YKVJ CPF
∃Z Z Z CU GZKUVGPEG CZKQOU $QQNQU UGV VJGQT[ 6JGP D[ VJG UCOG TGCUQPKPI CU YKVJ
(KPG UGV VJGQT[ CDQXG YG JCXG VJG HQNNQYKPI
$QQNQUOU .KVVNG 6JGQTGO $QQNQU UGV VJGQT[ CPF UGEQPFQTFGT 2GCPQ CTKVJOGVKE CTG KPVGTRTGVCDNG KP GCEJ QVJGT
O
8QP 0GWOCPP U CZKQOCVK\CVKQP QH UGV VJGQT[ WUGF RTGFKECVKXG F[CFKE UGEQPFQTFGT NQIKE DWV QVJGTYKUG YCU $QQNQU UGV VJGQT[ YKVJ C HGY CFFKVKQPCN UGVGZKUVGPEG CZKQOU 9KVJ CP KPIGPKQWU VGEJPKECN KORTQXGOGPV D[ .GX[ VJG PWODGT QH CFFKVKQPCN CZKQOU PGGFGF VQ IGV HWNNUVTGPIVJ UGV VJGQT[ ECP DG TGFWEGF VQ VYQ KPHKPKV[ CPF RQYGT UGV #FFKPI KPHKPKV[ IKXGU C VJGQT[ QH VJG UCOG UVTGPIVJ CU <( QT 2 CPF
CFFKPI CNUQ RQYGT UGV IKXGU C VJGQT[ QH VJG UCOG UVTGPIVJ CU <(% $WV YG FQ PGGF VJQUG VYQ HWTVJGT CZKQOU 6JG NQPI CPF UJQTV QH KV KU VJCV VJG XGTUKQPU QH VJG NKOKVCVKQP QH UK\G KFGC EQPUKFGTGF UQ HCT FQ PQV IGV QPG XGT[ HCT KPVQ VJG TGCNO QH UGV VJGQT[ DG[QPF CPCN[UKU 6JG RTQDNGO KU VJCV VJG RTKPEKRNG VJCV C UGV GZKUVU WPNGUU KVU GNGOGPVU YQWNF DG CU OCP[ CU VJGTG CTG QDLGEVU FQGU PQV VGNN WU VJCV C UGV GZKUVU WPNGUU KV KU XGT[ XGT[ NCTIG YKVJQWV VJG HWTVJGT CUUWORVKQP VJCV VJGTG CTG XGT[ XGT[ OCP[ QDLGEVU 6Q IGV OQTG QPG JCU VQ CRRGCN TGRGCVGFN[ VQ HTGUJ KPVWKVKQPU CDQWV JQY OCP[ KU VQQ OCP[ HQT VJG CZKQO QH KPHKPKV[ VQ VJG KPVWKVKQP VJCV KPHKPKVGN[ OCP[ KU PQV VQQ OCP[ HQT VJG RQYGT UGV CZKQO VQ CP KPVWKVKQP KORN[KPI VJCV WPEQWPVCDN[ OCP[ KU PQV VQQ OCP[ CPF KH QPG YCPVU KPCEEGUUKDNG ECTFKPCNU VQ CP KPVWKVKQP VJCV KPCEEGUUKDN[ OCP[ KU PQV VQQ OCP[ 6JG UGV VJGQTKUV YJQ TGNKGU QP VJG XGTUKQPU QH VJG NKOKVCVKQP QH UK\G KFGC EQPUKFGTGF UQ HCT JCU VQ MGGR EQOKPI DCEM VQ CUM HQT OQTG NKMG VJG HKUJGTOCP KP VJG HCKT[ VCNG CDQWV VJG OCIKE HNQWPFGT 6JG HKUJGTOCP HKPCNN[ CUMGF HQT VQQ OWEJ CU FKF UGV VJGQTKUVU YJGP VJG[ CUMGF HQT 4GKPJCTFV ECTFKPCNU +H QPG KU IQKPI VQ IGV QWV QH VJG NKOKVCVKQP QH UK\G KFGC C UKPING WPKHKGF HQTOCN UGVGZKUVGPEG CZKQO UWHHKEKGPV VQ IKXG C TGCNN[ RQYGTHWN UGV VJGQT[ QPG KU IQKPI VQ JCXG VQ CFQRV C FKHHGTGPV OGVJQF
6JG4GHNGEVKQP2TKPEKRNG
#NN VJG VJGQTKGU EQPUKFGTGF UQ HCT KP VJKU EJCRVGT JCXG OCFG WUG QH F[CFKE UGEQPFQTFGT NQIKE CPF VJG PQVKQP QH GSWKPWOGTQUKV[ .GV WU PQY OCMG C ENGCP DTGCM CPF JGPEGHQTVJ UVKEM YKVJ OQPCFKE UGEQPFQTFGT NQIKE +P RCTVKEWNCT YG IKXG WR CP[ SWCPVKVCVKXG XGTUKQP QH VJG NKOKVCVKQP QH UK\G KFGC 9JCV KU NGHV QH QWT (TGIGCP HTCOGYQTM HQT UGV VJGQT[ KU OQPCFKE UGEQPFQTFGT NQIKE YKVJ U[ODQNU e CPF ² CPF
↔ ∃: e Z: Z ∈ [ ↔ ∃; e [; ;Z
e Z: e [;
∈ YKVJ VJG CZKQOU QH UWDQTFKPCVKQP CPF GZVGPUKQPCNKV[
²Z
→ Z [ ↔ : ≡ ;
9JCV KU CNUQ NGHV KU VJG PQPSWCPVKVCVKXG XGTUKQP QH VJG NKOKVCVKQP QH UK\G KFGC YJKEJ GXGP KP KVU PQPOCZKOCNKUV HQTO IKXGU VJG CZKQO QH UGRCTCVKQP
:
⊆ ; → ∃[ e [; → ∃Z e Z:
9JCV KU OKUUKPI KU C HWTVJGT UGVGZKUVGPEG CZKQO QH VJG HQTO
`∃Z e Z:
→ VJGTG CTG VQQ OCP[ Z HCNNKPI WPFGT : VQ HQTO C UGV
0QY QPG EQPEGRV VJCV YG MPQY ECPPQV JCXG CP GZVGPUKQP KU VJG WPKXGTUCN EQPEGRV #RRNKGF VQ VJKU ECUG VGNNU WU VJKU
VJGTG VQQ OCP[ QDLGEVU VQ HQTO C UGV
6JG YGCMPGUU QH VJG XQP 0GWOCPP CRRTQCEJ YCU VJCV KV KP GHHGEV FGHKPGF
QVQQ OCP[ VQ
R VQ OGCP QCU OCP[ CU VJGTG CTG QDLGEVUR VJWU FGRTKXKPI QH CP[ UGTKQWU UWDUVCPEG 9JCV YG PGGF KU UQOG YC[ QH WPFGTUVCPFKPI QVQQ OCP[ VQ HQTO C UGVR VJCV YKNN HQTO C UGV
NGCXG YKVJ UQOG WUCDNG EQPVGPV 0QY VJGTG KU C YQTF VJCV CRRGCTU KP VJKU EQPPGEVKQP KP YTKVGTU CU FKXGTUG CU &WOOGVV CPF %CPVQT VJG YQTF
QKPFGHKPKVGR QT QWPDGUVKOOVR 6JGTG KU PQ
UGV QH CNN QDLGEVU CEEQTFKPI VQ DQVJ VJKPMGTU DGECWUG VJGTG CTG KPFGHKPKVGN[ OCP[ QDLGEVU &WOOGVV YJQ WUGU VJG GZRTGUUKQP
QKPFGHKPKVGN[ GZVGPUKDNGR KU UQQP NGF D[ JKU
KPVWKVKQPKUVKE RTQENKXKVKGU VQ WPFGTUVCPF KPFGHKPKVGPGUU CU DGKPI QH C OCVVGT QH UQOG UGPVGPEGU NCEMKPI VTWVJ XCNWGU %CPVQT YJQ WUGU VJG GZRTGUUKQP
QSWCPVKVCVKXGN[
R QT QSWCPVKVCVKX WPDGUVKOOVR WPFGTUVCPFU KPFGHKPKVGPGUU KP VJG UGPUG DGKPI
KPFGHKPKVG
KPECRCDNG QH DGKPI CUUKIPGF QPG QH JKU CNGRJU CU ECTFKPCNKV[ + RTQRQUG VQ WPFGTUVCPF
QKPFGHKPKVGPGUUR KP C OWEJ OQTG KOOGFKCVG YC[ CU QWPFGHKPCDKNKV[R VJWU
VJGTG CTG WPFGHKPCDN[ OCP[ QDLGEVU
#PF YJCV FQGU VJCV OGCP! 9GNN KV ECPPQV OGCP VJCV KV KU KORQUUKDNG VQ OCMG C VTWG UVCVGOGPV CDQWV JQY OCP[ QDLGEVU VJGTG CTG UKPEG KV KU VTWG UVCVGOGPV VJCV VJGTG KU CV NGCUV QPG QDLGEV HQT KPUVCPEG 9JCV + VCMG VQ OGCP KU VJCV VJQWIJ KV KU RQUUKDNG VQ OCMG C VTWG UVCVGOGPV CDQWV JQY OCP[ QDLGEVU VJGTG CTG VJGTG CTG VQQ OCP[ QDLGEVU HQT KV VQ DG RQUUKDNG HQT UWEJ C UVCVGOGPV VQ DG FGHKPKVKXG QH JQY OCP[ VJGTG CTG VJGTG YKNN PGEGUUCTKN[ DG PQV OGTGN[ CU OCP[ CU KU UCKF DWV OQTG CNUQ 9JCVGXGT QPG UC[U VJCV EQWNF UVKNN DG VTWG KH KV YGTG UCKF PQV CDQWV CNN QDLGEVU DWV CDQWV LWUV UQOG QDLGEVU HGYGT VJCP CNN
CP[VJKPI VJCV KU VTWG YJGP UCKF CDQWV CNN QDLGEVU TGOCKPU VTWG YJGP UCKF CDQWV LWUV UQOG QDLGEVU HGYGT VJCP CNN
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
W $WV
WPEQWPVCDN[ OCP[ QDLGEVU $WV D[ CICKP GXGP VJCV KU CP WPFGTUVCVGOGPV UQ
CNTGCF[ YG JCXG IQV HWTVJGT VJCP YG GXGT IQV DGHQTG #PF YJCV KU OGCPV D[ QHGYGT VJCP CNNR JGTG! *CXKPI FKUECTFGF F[CFKE UGEQPFQTFGT NQIKE CPF SWCPVKVCVKXG XGTUKQPU QH NKOKVCVKQP QH UK\G YG UVKNN JCXG VJG UQ HCT WPWUGF PQPSWCPVKVCVKXG DWV OCZKOCNKUV XGTUKQP CEEQTFKPI VQ YJKEJ QHGYGT VJCP CNNR DGEQOGU QHGY GPQWIJ VQ HQTO C UGVR 6JKU VTCPUHQTOU KPVQ VJG HQNNQYKPI
CP[VJKPI VJCV KU VTWG YJGP UCKF CDQWV CNN QDLGEVU TGOCKPU VTWG YJGP UCKF CDQWV LWUV VJG GNGOGPVU QH UQOG UGV
#V VJKU RQKPV JQYGXGT RCTCFQZ OC[ UGGO VQ VJTGCVGP UKPEG QPG VJKPI VJCV KU VTWG CDQWV CNN QDLGEVU KU VJCV VJG[ CTG PQV HGY GPQWIJ VQ HQTO C UGV CPF VJCV EGTVCKPN[ KUPOV VTWG QH VJG GNGOGPVU QH CP[ UGV V $WV VJG RCTCFQZ ECP DG TGUQNXGF D[ DGKPI ECTGHWN CDQWV QCDQWVR 6JG UVCVGOGPV Q6JGTG KU PQ UGV VJCV JCU CNN QDLGEVU CU GNGOGPVUR KU VTWG CPF VJG UVCVGOGPV Q6JGTG KU PQ UGV VJCV JCU CNN GNGOGPVU QH V CU GNGOGPVUR KU HCNUG DWV VJG HCNUG UVCVGOGPV KUPOV LWUV CDQWV VJG GNGOGPVU QH V UKPEG VJG WPKXGTUCN PGICVKXG QVJGTG KU PQ UGVWR UC[U UQOGVJKPI CDQWV CNN UGVU PQV LWUV VJQUG VJCV CTG GNGOGPVU QH V 6JG UVCVGOGPV KP VJG XKEKPKV[ VJCV KU LWUV CDQWV VJG GNGOGPVU QH V KU TCVJGT VJG UVCVGOGPV Q6JGTG KU PQ UGV COQPI VJG GNGOGPVU QH V VJCV JCU CNN GNGOGPVU QH V CU GNGOGPVUR CPF VJCV KU PQV RCTCFQZKECN %NGCTN[ VJG PQVKQP TGSWKTGF JGTG KU VJCV QH TGNCVKXK\CVKQP QH SWCPVKHKGTU 1WT IGPGTCN PQVCVKQP HQT TGNCVKXK\KPI VJG SWCPVKHKGTU KP C HQTOWNC φ VQ C HQTOWNC θ Z TGRNCEKPI ∀Z W D[ ∀Z θ Z → W CPF UKOKNCTN[ HQT ∃ JCU DGGP φθ 9JGP θ Z KU Z ∈ V UQ YG CTG TGRNCEKPI ∀Z W D[ ∀Z Z ∈ V → W CPF UKOKNCTN[ HQT ∃ YG LWUV YTKVG φ 5KPEG YG CTG WUKPI UGEQPFQTFGT NQIKE YG PGGF VQ UC[ JQY UGEQPFQTFGT SWCPVKHKGTU CTG VQ DG TGNCVKXK\GF 4GNCVKXK\KPI ∀: DGEQOGU V
∀: ∀Z :Z → θ Z →
W
QT
∀: ∀Z :Z → Z ∈ V →
W
KP VJG IGPGTCN QT VJG URGEKCN ECUG TGURGEVKXGN[ CPF UKOKNCTN[ HQT ∃ 6JG HQTO QP VJG TKIJV OC[ DG CDDTGXKCVGF ∀: : ⊆ V → W QT GXGP LWUV ∀: ⊆ V W *CXKPI VJKU PQVKQP QH TGNCVKXK\CVKQP KP JCPF YG ECP UC[ VJCV YJCV KU OGCPV D[ KU VJG HQNNQYKPI
φ → ∃V φV
+V KU PQV ENCKOGF VJCV KU UGNHGXKFGPVN[ VTWG QT GXGP UGNHGXKFGPVN[ EQPUKUVGPV $[ )ÑFGNOU UGEQPF KPEQORNGVGPGUU VJGQTGO C UGNHGXKFGPVN[ EQPUKUVGPV VJGQT[ YQWNF JCXG VQ DG XGT[ YGCM KPFGGF 4CVJGT KU UQOGVJKPI VJCV JCU DGGP CTTKXGF CV D[ HQNNQYKPI C EGTVCKP JGWTKUVKE VTCKP QH VJQWIJV UVCTVKPI YKVJ VJG KFGC QH NKOKVCVKQP QH UK\G CPF VJG CFFKVKQPCN UWIIGUVKQP VJCV QVQQ OCP[R UJQWNF DG WPFGTUVQQF VQ OGCP QKPFGHKPKVGN[ QT KPFGHKPCDN[ OCP[R TCVJGT VJCP VQ OGCP QGSWKPWOGTQWU YKVJ CNN QDLGEVUR CU KP RTGXKQWU CVVGORVU 2TKPEKRNGU QH VJG IGPGTCN UJCRG QH UC[KPI VJCV YJCVGXGT JQNFU KP VJG OCETQEQUO QH CNN QDLGEVU JQNFU KP VJG OKETQEQUO QH GNGOGPVU QH UQOG UGV V CTG ECNNGF TGHNGEVKQP RTKPEKRNGU 4GHNGEVKQP RTKPEKRNGU YGTG KPVTQFWEGF KPVQ CZKQOCVKE UGV VJGQT[ D[ #\TKGN .GX[ CPF GURGEKCNN[ FGXGNQRGF D[ 2CWN $GTPC[U YJQ UJQYGF VJG[ EQWNF DG QH ITGCV RQYGT 0GKVJGT .GX[ PQT $GTPC[U KU FKTGEVN[ EQPEGTPGF YKVJ OQVKXCVKPI TGHNGEVKQP RTKPEKRNGU CU CZKQOU .GX[ FGTKXGU QPG HTQO VJG CZKQOU QH <( CPF $GTPC[U GZRNQTGU VJG EQPUGSWGPEGU QH VCMKPI QPG CU CP WPCTIWGF CUUWORVKQP 6JG TGHNGEVKQP RTKPEKRNG KU KPFGGF VCPVCNK\KPIN[ ENQUG VQ QPG UVWFKGF D[ $GTPC[U 6JG $GTPC[U RTKPEKRNG FKHHGTU HTQO QPN[ D[ KORQUKPI C EGTVCKP GZVTC VGEJPKECN EQPFKVKQP QP V 6JG VCUM QH IQKPI DCEM QXGT VJG JGWTKUVKE NKPG QH VJQWIJV NGCFKPI VQ CPF VT[KPI VQ OQVKXCVG VJG GZVTC VGEJPKECN EQPFKVKQP D[ URGEKCN RNGCFKPI UGGOU FCWPVKPI CPF HQTVWPCVGN[ KV KU WPPGEGUUCT[ UKPEG KV VWTPU QWV VQ DG RQUUKDNG VQ FGTKXG VJG $GTPC[U RTKPEKRNG HTQO WUKPI C VGEJPKSWG QH $GTPC[U JKOUGNH EQODKPGF YKVJ VJG QPG FKUVKPEVKXGN[ (TGIGCP HGCVWTG QH QWT CZKQOCVK\CVKQP VJG UWDQTFKPCVKQP CZKQOU YJKEJ JCXG PQV [GV DGGP WUGF 6JG $GTPC[U VGEJPKSWG CRRNKGF VQ KU UKORN[ VJKU VJCV KH YG JCXG RTQXGF C VJGQTGO θ VJGP CP[ φ KORNKGU θ φ CPF CRRN[KPI VQ VJKU EQPLWPEVKQP TCVJGT VJCP LWUV φ YG IGV φ → ∃V θ φ YJKEJ KU VJG UCOG CU φ → ∃V θ φ +H θ KORNKGU UQOG URGEKCN RTQRGTV[ ψ V VJGP YG IGV φ → ∃V ψ V φ +H VJG TGNCVKXK\CVKQP QH C VJGQTGO VQ V KORNKGU C URGEKCN RTQRGTV[ QH V VJGP VJG TGHNGEVKQP RTKPEKRNG ECP DG UVTGPIVJGPGF VQ TGSWKTG V VQ JCXG VJKU URGEKCN RTQRGTV[ 5KPEG VJG EQPLWPEVKQP QH CP[ PWODGT QH VJGQTGOU KU UVKNN C VJGQTGO VJG VJGQTGOTGNCVKXK\CVKQP RTKPEKRNG ECP DG WUGF QXGT CPF QXGT VQ IGV V
V
V
V
V
OQTG CPF OQTG URGEKCN RTQRGTVKGU IKXKPI OQTG CPF OQTG VJGQTGOU IKXKPI OQTG CPF OQTG URGEKCN RTQRGTVKGU C HGCVWTG KP EQPPGEV YKVJ YJKEJ $GTPC[U WUGU VJG GZRTGUUKQP QUGNHTGKPHQTEKPIR .GV WU HKTUV CRRN[ VJKU KFGC VCMKPI CU QWT θ VJG UWDQTFKPCVKQP CZKQO +VU TGNCVKXK\CVKQP TGCFU CU HQNNQYU ∀Z Z ∈ V → ²Z ↔ ∃: ⊆ V e Z:
6JG WUGHWN RCTV QH VJG DKEQPFKVKQPCN KU VJG KORNKECVKQP HTQO NGHV VQ TKIJV (QT UWRRQUG Z ∈ V CPF [ ∈ Z 6JGP VJKU FKTGEVKQP QH VJG DKEQPFKVKQPCN VGNNU WU VJGTG KU CP : ⊆ V UWEJ VJCV e Z: 6JG UWDQTFKPCVKQP CZKQO VGNNU WU VJCV UKPEG [ ∈ Z YG JCXG :[ CPF VJKU VQIGVJGT YKVJ : ⊆ V VGNNU WU YG JCXG [ ∈ V #P[ GNGOGPV QH CP GNGOGPV QH V KU CP GNGOGPV QH V #U YCU OGPVKQPGF KP VJG FKUEWUUKQP QH NCTIG ECTFKPCNU KP %JCRVGT C UGV YKVJ VJKU URGEKCN RTQRGTV[ KU ECNNGF VTCPUKVKXG 6JG URGEKCN RTQRGTV[ QH VTCPUKVKXKV[ KU VJG QGZVTC VGEJPKECN EQPFKVKQPR KP VJG $GTPC[U TGHNGEVKQP RTKPEKRNG YJKEJ COQWPVU VQ YJCV YG JCXG LWUV RTQXGF PCOGN[
φ → ∃V V VTCPUKVKXG φV
.GV OG ECNN VJG UGV VJGQT[ KP VJG OQPCFKE UGEQPFQTFGT NCPIWCIG YKVJ LWUV ² CPF ∈ CPF VJG CZKQOU QH GZVGPUKQPCNKV[ UGRCTCVKQP KP C UKPINGCZKQO UGEQPFQTFGT XGTUKQP CPF TGHNGEVKQP KP VJG UVTQPIGT HQTO $GTPC[U UGV VJGQT[ $ (QT VJQWIJ PQV RTGEKUGN[ VJG U[UVGO YKVJ YJKEJ $GTPC[U YQTMGF KV KU PGCTN[ GPQWIJ UQ HQT RTGUGPV RWTRQUGU #PF NGV OG ECNN VJG UGV VJGQT[ VJCV CFFU e CPF VJG UWDQTFKPCVKQP CZKQOU CPF UWDUVKVWVGU TGHNGEVKQP KP VJG YGCMGT HQTO (TGIGCPK\GF $GTPC[U UGV VJGQT[ ($ $GTPC[U KP GHHGEV RTQXGF VJCV CNN VJG TGOCKPKPI GZKUVGPEG CZKQOU QH UGEQPFQTFGT
+V JCU IGPGTCNN[ DGGP O[ RQNKE[ PQV VQ TGRTQFWEG RTQQHU CNTGCF[ TGCFKN[ CXCKNCDNG KP VJG NKVGTCVWTG GZEGRV YJGTG KV KU PGEGUUCT[ VQ TGXKGY VJG RTQQH KP QTFGT VQ QDVCKP TGHKPGOGPVU 6JG QPN[ TGHKPGOGPV + JCXG VQ QHHGT VQ VJG VJGQTGO QH $GTPC[U CRCTV HTQO VJG KPKVKCN JGWTKUVKE OQVKXCVKQP HQT VJG TGHNGEVKQP RTKPEKRNG KU VJG QDUGTXCVKQP VJCV VJG GZVTC VGEJPKECN EQPFKVKQP QH VTCPUKVKXKV[ FQGU PQV PGGF VQ DG CUUWOGF DWV ECP DG FGFWEGF WUKPI VJG (TGIGCP UWDQTFKPCVKQP CZKQOU CPF VJKU TGHKPGOGPV JCU CNTGCF[ DGGP RTGUGPVGF 6JG $GTPC[U RTQQH KU UQ DGCWVKHWN JQYGXGT VJCV KV KU FKHHKEWNV VQ TGHTCKP HTQO RTGUGPVKPI CV NGCUV RCTV QH KV CPF VQ FQ UQ OC[ HCEKNKVCVG WPFGTUVCPFKPI QH YJCV JCU IQPG DGHQTG 9G IQV C TGHKPGOGPV QH TGHNGEVKQP WUKPI VJG UWDQTFKPCVKQP CZKQOU CPF GZVGPUKQPCNKV[ DWV JCXG PQV [GV VTKGF TGHNGEVKPI VJG UGRCTCVKQP CZKQO *GTG QUGRCTCVKQPR OC[ DG CP[ EQPXGPKGPV UKPINGCZKQO HQTOWNCVKQP QH VJG RTKPEKRNG UKPEG CNN VJG XCTKQWU HQTOWNCVKQPU CTG VJGQTGOU 6JG OQUV EQPXGPKGPV VQ EQPUKFGT IKXGU VJG HQNNQYKPI CU KVU TGNCVKXK\CVKQP
∀Z ∈ V ∀: ⊆ V ∃[ ∈ V ∀\ ∈ V \ ∈ [ ↔ \ ∈ Z :\
*GTG V OC[ DG VCMGP VQ DG VTCPUKVKXG 0QY KH Z ∈ V CPF Y ⊆ Z HKTUV PQVG VJCV Z ⊆ V D[ VTCPUKVKXKV[ 5Q YG JCXG Y ⊆ V CPF VJGTGHQTG VJG EQPEGRV : n\ \ ∈ Y UCVKUHKGU : ⊆ V 5Q TGNCVKXK\GF UGRCTCVKQP VGNNU WU VJGTG KU C [ ∈ V HQT YJKEJ YG JCXG VJG HQNNQYKPI
∀\ ∈ V \ ∈ [ ↔ \ ∈ Z :\
6JG TGNCVKXK\CVKQP KP VJG SWCPVKHKGT ∀\ ∈ V KP KU TGFWPFCPV UKPEG VJG EQPFKVKQP \ ∈ [ QP VJG NGHV UKFG CPF VJG EQPFKVKQP \ ∈ Z QP VJG TKIJV UKFG QH VJG DKEQPFKVKQPCN KP DQVJ KORN[ \ ∈ V CNTGCF[ IKXGP VJCV Z ∈ V CPF [ ∈ V CPF V KU VTCPUKVKXG 5Q YG YJCV YG TGCNN[ JCXG KU VJG HQNNQYKPI
∀\ \ ∈ [ ↔ \ ∈ Z :\
$WV TGECNNKPI VJCV : n\ \ ∈ Y CPF Y ⊆ Z VJG EQPLWPEVKQP \ ∈ Z :\ QP VJG TKIJV UKFG QH VJG DKEQPFKVKQPCN KP KU GSWKXCNGPV VQ \ ∈ Y CPF UQ YG JCXG Y [ ∈ V +P QVJGT YQTFU GXGT[ UWDUGV Y QH CP GNGOGPV Z QH V KU CP GNGOGPV QH V #U YCU UCKF KP VJG FKUEWUUKQP QH NCTIG ECTFKPCNU KP %JCRVGT C VTCPUKVKXG UGV YKVJ VJKU URGEKCN RTQRGTV[ KU ECNNGF UWRGTVTCPUKVKXG 5Q YG JCXG GUVCDNKUJGF VJG HQNNQYKPI UWRGT$GTPC[U XGTUKQP QH TGHNGEVKQP
φ → ∃V V UWRGTVTCPUKVKXG φV
6Q FGTKXG RCKTKPI YG QPN[ PGGF VJG QTKIKPCN XGTUKQP QH TGHNGEVKQP )KXGP W CPF X YG TGHNGEV VJG HQNNQYKPI NQIKECN VTWVJ
∃Z Z W ∃[ [ X
VQ IGV C UGV V UWEJ VJCV JQNFU TGNCVKXK\GF VQ V YJKEJ KU VQ UC[ VJG HQNNQYKPI JQNFU
∃Z ∈ V Z W ∃[ ∈ V [ X
$WV KU LWUV C NQPIYKPFGF YC[ QH UC[KPI W ∈ V CPF X ∈ V 9G OC[ PQY CRRN[ UGRCTCVKQP VQ UGRCTCVG QWV HTQO V LWUV VJQUG QH KVU GNGOGPVU VJCV CTG GKVJGT KFGPVKECN YKVJ W QT KFGPVKECN YKVJ X CPF UQ QDVCKP VJG RCKT ]W X_ (QT WPKQP YG WUG VJG $GTPC[U XGTUKQP YKVJ VTCPUKVKXKV[ VQ UJQY VJCV HQT GXGT[ W VJGTG KU C VTCPUKVKXG UGV VQ YJKEJ KV DGNQPIU 5WEJ C V YKNN EQPVCKP CNN VJG GNGOGPVU QH W CPF CNN VJG GNGOGPVU QH VJQUG GNGOGPVU CPF YG ECP UGRCTCVG QWV VJG GNGOGPVU QH GNGOGPVU VQ IGV VJG WPKQP ∪W (QT RQYGT YG WUG VJG XGTUKQP YKVJ UWRGTVTCPUKVKX[ VQ UJQY VJCV HQT GXGT[ W VJGTG KU C UWRGTVTCPUKVKXG CPF VTCPUKVKXG UGV V YKVJ W ∈ V 5WEJ C V YKNN EQPVCKP CNN VJG UWDUGVU QH W CPF YG ECP UGRCTCVG QWV VJQUG UWDUGVU VQ IGV VJG RQYGT UGV ℘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
²Z ²[ ∀\ \ ∈ Z ↔ \ ∈ [ → Z [
∀\ \ ∈ Z ↔ \ ∈ [ → Z [
1WT HQTO KU VJG QPN[ TGCUQPCDNG HQTO KH QPG CUUWOGU CU (TGIG JKOUGNH CNYC[U FKF VJCV VJG QDLGEV XCTKCDNGU TCPIG QXGT CDUQNWVGN[ CNN QDLGEVU 6JG WUWCN CFQRVKQP QH HQTO
KU PQV CP GZRTGUUKQP QH VJG WPTGCUQPCDNG DGNKGH VJCV CDUQNWVGN[ GXGT[ QDLGEV KU C UGV DWV TCVJGT QH CP KPVGPVKQP VQ KORQUG CP WP(TGIGCP TGUVTKEVKQP QP VJG TCPIG QH VJG QDLGEV XCTKCDNGU 6JG[ CTG VQ DG TGUVTKEVGF VQ UGVU CPF PQV LWUV VQ UGVU DWV VQ UGVU QH YJKEJ VJG GNGOGPVU CTG CNN QH VJGO UGVU CPF CTG PQV LWUV UGVU DWV CTG UGVU YJQUG GNGOGPVU CTG CNN QH VJGO UGVU CPF CTG PQV LWUVW +P UJQTV VJG TGUVTKEVKQP KU VQ UGVU Z UWEJ VJCV GXGT[ QDLGEV UVCPFKPI KP VJG CPEGUVTCN QH VJG GNGOGPVJQQF TGNCVKQP VQ Z KU C UGV 5WEJ CTG ECNNGF RWTG UGVU 6JG QDXKQWU UVTCVGI[ HQT IGVVKPI VJG WUWCN XGTUKQP QH GZVGPUKQPCNKV[ YQWNF DG VQ TGNCVKXK\G SWCPVKHKGTU VQ RWTG UGVU 6Q FQ VJKU YG OWUV HKTUV HKPF C HQTOWNC θ Z GZRTGUUKPI VJCV Z KU RWTG 6JKU KU PQ RTQDNGO WUKPI UGEQPFQTFGT NQIKE YJGTG YG JCXG (TGIGOU FGHKPKVKQP QH VJG CPEGUVTCN $WV KP HCEV VJGTG KU CP GSWKXCNGPV HKTUVQTFGT θ Z YJKEJ UKORN[ UC[U VJCV VJGTG KU C VTCPUKVKXG V UWEJ VJCV Z ∈ V CPF GXGT[ GNGOGPV QH V KU C UGV 5KPEG C VTCPUKVKXG V UWEJ VJCV Z ∈ V YKNN JCXG CNN QDLGEVU UVCPFKPI KP VJG CPEGUVTCN QH VJG GNGOGPVJQQF TGNCVKQP VQ Z CU GNGOGPVU KH CNN GNGOGPVU QH V CTG UGVU Z KU EGTVCKPN[ RWTG %QPXGTUGN[ YG MPQY VJCV GXGT[ Z KU CP GNGOGPV QH UQOG VTCPUKVKXG UGV V CPF YG ECP UGRCTCVG QWV HTQO V LWUV Z CPF VJG QDLGEVU VJCV UVCPF KP VJG CPEGUVTCN QH VJG GNGOGPVJQQF TGNCVKQP VQ Z VQ HQTO C UGV W VJCV YKNN DG VTCPUKVKXG CPF YKNN JCXG Z ∈ W CPF VJCV KH Z KU RWTG YKNN JCXG QPN[ UGVU CU GNGOGPVU 9G PGGF VQ EJGEM VJCV KP IGVVKPI GZVGPUKQPCNKV[ KP XGTUKQP YG JCXG PQV NQUV CP[VJKPI GNUG 9G PGGF VQ EJGEM HQT KPUVCPEG VJCV VJG CZKQO QH RCKTKPI UVKNN JQNFU YJGP SWCPVKHKGTU CTG TGNCVKXK\GF VQ RWTG UGVU +H VJGTG GZKUV VTCPUKVKXG V CPF U EQPVCKPKPI W CPF X TGURGEVKXGN[ CPF DQVJ JCXKPI QPN[ UGVU CU GNGOGPVU YG PGGF VQ UJQY VJGTG KU C VTCPUKVKXG T EQPVCKPKPI Y ]W X_ CPF JCXKPI QPN[ UGVU CU GNGOGPVU 6JKU ECP DG FQPG D[ XGTKH[KPI VJCV T U ∪ V ∪ ]Y_ YKNN FQ DWV VJGP VJG UCOG JCU VQ DG FQPG HQT GXGT[ QVJGT UGVGZKUVGPEG CZKQO /QTGQXGT VQ IGV VJG CZKQO QH HQWPFCVKQP QPG PGGFU VQ FQ UQOGVJKPI UKOKNCT YKVJ C UGEQPF TQWPF QH XGTKHKECVKQPU (QTVWPCVGN[ VJG FGVCKNU CTG IKXGP KP KPVTQFWEVQT[ VGZVDQQMU CPF PGGF PQV EQPEGTP WU JGTG #U HQT EJQKEG YJKEJ JCU OCP[ GSWKXCNGPVU UWEJ CU VJG RTKPEKRNG VJCV GXGT[ UGV ECP DG YGNNQTFGTGF VJGTG CTG UGXGTCN QRVKQPU 6JG HKTUV QRVKQP KU VQ QDVCKP EJQKEG D[ TGNCVKXK\CVKQP VQ )ÑFGNOU EQPUVTWEVKDNG UGVU M VJKU KU JQY )ÑFGN UJQYGF <(% ECP DG KPVGTRTGVGF KP <( 6JG RTQEGFWTG KU CTVKHKEKCN UKPEG KP EQPVTCUV VQ VJG ECUG QH GZVGPUKQPCNKV[ VJQUG CFQRVKPI #% CTG PQV GZRTGUUKPI CP KPVGPVKQP VQ NKOKV VJG TCPIG QH VJG QDLGEV XCTKCDNGU VQ EQPUVTWEVKDNG UGVU $WV CTVKHKEKCNKV[ KU PQV CNYC[U CP QDLGEVKQP 1PG
UJQWNF DG CYCTG VJQWIJ VJCV KH QPG KU RTGRCTGF VQ IGV CZKQOU D[ TGNCVKXK\CVKQP VJGP CP QUVGPUKDN[ OWEJ OQTG QOKPKOCNKUVKER UVCTVKPI RQKPV VJCP ($ YKNN IGV WU VQ VJG UCOG IQCN 6JKU JCU DGGP GUVCDNKUJGF KP TGEGPV YQTM QH (TKGFOCP ! YJKEJ + YKNN FKITGUU VQ FGUETKDG DTKGHN[ (TKGFOCP HKTUV EQPUKFGTU C NCPIWCIG YKVJ QPN[ QDLGEV XCTKCDNGU CPF QPN[ ∈ PQ UGEQPFQTFGT CRRCTCVWU CV CNN CPF PQV GXGP *G EQPUKFGTU C HKTUV VJGQT[ + YKNN ECNN *(4 HQT *CTXG[ (TKGFOCP TGHNGEVKXG UGV VJGQT[ HKTUV XGTUKQP JCXKPI PQ CZKQOU GZEGRV VJG HQNNQYKPI UEJGOG φθ → ∃V ∀Z Z ∈ V → θ Z φV
+H φ JQNFU TGNCVKXK\GF VQ θ VJGP VJGTG KU CP V UWEJ VJCV θ JQNFU HQT CNN GNGOGPVU QH V CPF φ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εU[ODQN YJKEJ KP GHHGEV DWKNFU #% KPVQ HKTUVQTFGT NQIKE 6Q FGUETKDG VJKU QRVKQP KP FGVCKN YQWNF VCMG WU VQQ HCT CHKGNF # HQWTVJ QRVKQP KU VQ FQ YJCV KU FQPG KP OQUV EQPXGPVKQPCN CZKQOCVK\CVKQPU CPF UKORN[ CFQRV #% CU CP CZKQO VJCV JCU VQ DG KPFGRGPFGPVN[ OQVKXCVGF K
5GEQPF1TFGT.QIKE4GEQPUKFGTGF
4CVJGT VJCP GNCDQTCVG HWTVJGT QP VJG UGVVJGQTGVKE OCVGTKCN + JCXG CNNWFGF VQ OQUV QH YJKEJ KU TGCFKN[ CXCKNCDNG GNUGYJGTG CPF XGT[ NKVVNG QH YJKEJ JCU OWEJ UCXQTKPI QH (TGIG CDQWV KV + YKNN WUG VJG TGOCKPKPI URCEG CXCKNCDNG VQ OG VQ ECUV C OQTG ETKVKECN G[G WRQP VJG VQQN GORNQ[GF KP QPG YC[ QT CPQVJGT KP CNN PGQ(TGIGCP RTQLGEVU UGEQPFQTFGT NQIKE KVUGNH 9KVJ HKTUVQTFGT NQIKE YG JCXG C U[PVCEVKE QT RTQQHVJGQTGVKE PQVKQP QH C EQPENWUKQP RGTJCRU C EQPLGEVWTGF VJGQTGO DGKPI FGFWEKDNG HTQO C RTGOKUU RGTJCRU C EQPLWPEVKQP QH CZKQOU QH UQOG VJGQT[ 5RGEKCN QT FGIGPGTCVG ECUGU IKXG WU VJG PQVKQPU QH KPEQPUKUVGPE[ CPF FGOQPUVTCDKNKV[ C RTGOKUU KU KPEQPUKUVGPV KH CPF QPN[ KH CP[ EQPENWUKQP KU FGFWEKDNG HTQO KV CPF C EQPENWUKQP KU FGOQPUVTCDNG KH KV KU FGFWEKDNG HTQO CP[ RTGOKUU +H YG UVCTV YKVJ VJG PQVKQP QH KPEQPUKUVGPE[ YG ECP CNVGTPCVKXGN[ FGHKPG FGFWEKDKNKV[ QH ψ HTQO φ CU VJG KPEQPUKUVGPE[ QH `ψ φ CPF FGOQPUVTCDKNKV[ QH ψ CU VJG KPEQPUKUVGPE[ QH ` ψ 9G CNUQ JCXG C UGOCPVKE QT OQFGNVJGQTGVKE PQVKQP QH C EQPENWUKQP DGKPI C EQPUGSWGPEG QH C RTGOKUU ψ KU C EQPUGSWGPEG QH φ KH CPF QPN[ KH GXGT[ OQFGN VJCV OCMGU φ VTWG OCMGU ψ VTWG 9G CNUQ JCXG TGNCVGF PQVKQPU QH WPUCVKUHKCDKNKV[ CPF XCNKFKV[ C RTGOKUU KU WPUCVKUHKCDNG KH VTWG KP PQ OQFGN CPF C EQPENWUKQP KU XCNKF KH VTWG KP CNN OQFGNU +H YG UVCTV YG VJG PQVKQP QH WPUCVKUHKCDKNKV[ YG ECP CNVGTPCVKXGN[ FGHKPG ψ DGKPI C EQPUGSWGPEG QH φ CU VJG WPUCVKUHKCDKNKV[ QH `ψ φ CPF XCNKFKV[ QH ψ CU VJG WPUCVKUHKCDKNKV[ QH `ψ 6JG VYQ VTKRNGU QH PQVKQPU EQKPEKFG D[ VJG UQWPFPGUU VJGQTGO CPF VJG )ÑFGN EQORNGVGPGUU VJGQTGO θ KU UCVKUHKCDNG KH CPF QPN[ KH θ KU EQPUKUVGPV (TQO VJKU KV HQNNQYU CNUQ VJCV FGFWEKDKNKV[ EQKPEKFGU YKVJ EQPUGSWGPEG CPF FGOQPUVTCDKNKV[ YKVJ XCNKFKV[ 6JG EQPPGEVKQP DGVYGGP VJG U[PVCEVKE CPF UGOCPVKE PQVKQPU OGCPU VJCV OQFGNVJGQTGVKE TGUWNVU ECP DG WUGF VQ FGTKXG RTQQHVJGQTGVKE EQTQNNCTKGU CPF XKEG XGTUC 6JWU HCT + JCXG EQPUKFGTGF QPN[ U[PVCEVKE PQVKQPU KP EQPPGEVKQP YKVJ UGEQPFQTFGT VJGQTKGU CPF JCXG GORJCUK\GF VJCV UQ HCT CU EQPUKUVGPE[ FGFWEKDKNKV[ CPF FGOQPUVTCDKNKV[ CTG EQPEGTPGF UGEQPFQTFGT NQIKE KU LWUV OCP[UQTVGF HKTUVQTFGT NQIKE CPF CNN VJG VQQNU WUGF KP EQPPGEVKQP HQT VJG NCVVGT ECP DG WUGF HQT VJG HQTOGT 6JKU KPENWFGU VJG WUG QH OQFGNU KP VJG UGPUG QH HKTUVQTFGT NQIKE VQ GUVCDNKUJ EQPUKUVGPE[ 5QOGVKOGU VJGTG CTG CFXCPVCIGU VQ GUVCDNKUJKPI EQPUKUVGPE[ RTQQHVJGQTGVKECNN[ TCVJGT VJCP OQFGNVJGQTGVKECNN[ DGECWUG C RTQQHVJGQTGVKE CTIWOGPV HQT EQPUKUVGPE[ V[RKECNN[ IKXGU CFFKVKQPCN KPHQTOCVKQP DG[QPF OGTG EQPUKUVGPE[ KP C YC[ C OQFGNVJGQTGVKE CTIWOGPV V[RKECNN[ FQGU PQV $WV C OQFGNVJGQTGVKE RTQQH QH EQPUKUVGPE[ FQGU RTQXG EQPUKUVGPE[
CPF C VJGQT[ FQGUPOV DGEQOG CP[ OQTG EQPUKUVGPV VJCP KV CNTGCF[ YCU YJGP C OQFGNVJGQTGVKE RTQQH KU TGRNCEGF D[ C RTQQHVJGQTGVKE RTQQH +P YQTMKPI YKVJ OQFGNU HQT UGEQPFQTFGT NQIKE KV KU CNYC[U UWHHKEKGPV VQ EQPUKFGT QPN[ ∈OQFGNU QT OQFGNU YJGTG VJG FQOCKP QH VJG UGEQPFQTFGT XCTKCDNGU EQPUKUVU QH UWDUGVU QH VJG FQOCKP QH VJG HKTUVQTFGT XCTKCDNGU UKPEG KV KU CP GCU[ VJGQTGO VJCV CP CTDKVTCT[ OQFGN KU KUQOQTRJKE VQ CP ∈OQFGN UQ NQPI CU QPG KU CUUWOKPI GZVGPUKQPCNKV[ CU + CNYC[U JCXG DGGP 5QOGVKOGU VJG FQOCKP QH VJG UGEQPFQTFGT XCTKCDNGU EQPUKUVU QH CNN UWDUGVU QH VJG FQOCKP QH VJG HKTUVQTFGT XCTKCDNGU QHVGP KV FQGU PQV +P VJG HQTOGT ECUG YG JCXG C UVCPFCTF QT 6CTUMK OQFGN KP VJG NCVVGT ECUG C IGPGTCN QT *GPMKP OQFGN )GPGTCN OQFGNU CTG LWUV CU IQQF CU UVCPFCTF OQFGNU HQT GUVCDNKUJKPI HCEVU CDQWV U[PVCEVKE EQPUKUVGPE[ FGFWEKDKNKV[ QT FGOQPUVTCDKNKV[ 6JGTG KU CNUQ JQYGXGT C PQVKQP QH UVCPFCTF UGOCPVKE EQPUGSWGPEG QH YJKEJ + JCXG UCKF PQVJKPI UQ HCT CEEQTFKPI VQ YJKEJ ψ KU C UVCPFCTF EQPUGSWGPEG QH φ KH CPF QPN[ KH ψ KU VTWG KP GXGT[ UVCPFCTF OQFGN KP YJKEJ φ KU VTWG 9KVJ VJKU PQVKQP EQOG CUUQEKCVGF PQVKQPU QH UVCPFCTF WPUCVKUHKCDKNKV[ QT UCVKUHKCDKNKV[ CPF UVCPFCTF XCNKFKV[ QT KPXCNKFKV[ #PF KP CITGGOGPV YKVJ VJG UQWPFPGUU VJGQTGO CPF KP EQPVTCUV VQ VJG )ÑFGN KPEQORNGVGPGUU VJGQTGO HQT HKTUVQTFGT NQIKE KV KU MPQYP VJCV UVCPFCTF EQPUGSWGPEG KU C
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RTQQH VJGQT[ CPF OQFGN VJGQT[ QPG KU KPHQTOCNN[ YQTMKPI KP UQOG MKPF QH DCEMITQWPF OCVJGOCVKECN OGVCVJGQT[ #PF YJGP FQKPI OQFGN VJGQT[ IGPGTCNN[ VJKU OGVCVJGQT[ KU UQOG MKPF QH UGV VJGQT[ VJQWIJ YJGP FQKPI QPN[ RTQQH VJGQT[ KV OC[ DG UQOGVJKPI OWEJ YGCMGT GXGP CU YGCM CU 3 +∆ UWRGTGZR QT YGCMGT 6JG FGHCWNV CUUWORVKQP YJGP PQ OGVCVJGQT[ KU GZRNKEKVN[ KPFKECVGF KU VJCV QPG KU EQPHQTOKPI VQ IGPGTCNN[ CEEGRVGF UVCPFCTFU KP EQPVGORQTCT[ OCVJGOCVKEU YJKEJ KU VQ UC[ VJCV QPG KU YQTMKPI YKVJKP VJG IGPGTCNN[ CEEGRVGF UGV VJGQT[ QH EQPVGORQTCT[ OCVJGOCVKEU <(% VJQWIJ QHVGP QPG KU TGCNN[ WUKPI HCT NGUU VJCP VJG HWNN UVTGPIVJ VJGTGQH +V VWTPU QWV VJCV KV OCMGU C TGCN FKHHGTGPEG VQ VJG SWGUVKQP QH VJG TGNCVKQPUJKR DGVYGGP UVCPFCTF UCVKUHKCDKNKV[ CPF EQPUGSWGPEG CPF XCNKFKV[ CPF KPVWKVKXG UCVKUHKCDKNKV[ CPF EQPUGSWGPEG CPF XCNKFKV[ LWUV YJCV UGV VJGQT[ QPG KU YQTMKPI KP CPF VJG UGV VJGQT[ ($ KU GZCEVN[ YJCV KU PGGFGF VQ DG CDNG VQ IKXG C FGEKUKXG CPUYGT VQ VJG SWGUVKQP QH VJG TGNCVKQP QH UVCPFCTF VQ KPVWKVKXG
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∃Z∃T T YGNNQTFGTU Z
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WPKSWGPGUU ENCKOU VJGP JCXG VQ DG GUVCDNKUJGF VQ EQORNGVG VJG RTQQH 6JCV C PQPVTKXKCN COQWPV QH PWODGT VJGQT[ ECP KP C UGPUG DG FQPG GXGP YKVJQWV VJG GZKUVGPEG QH PWODGTU KU GORJCUK\GF KP C UGTKGU QH RCRGTU D[ #PFTGY $QWEJGT VJCV JCXG CU QH VJG VKOG QH VJKU YTKVKPI PQV [GV DGGP RWDNKUJGF CPF VJCV ECOG VQ O[ CVVGPVKQP VQQ NCVG VQ DG OCFG WUG QH JGTG 2TGRTKPVU CTG CXCKNCDNG CV YYYCPFTGYDQWEJGTEQO 6JKU OCMGU C YQWNF TGSWKTG
VJKTFQTFGT
UWHHKEKGPV
EQPFKVKQP HQT PQPKPHNCVKPI 6JG
NQIKE CPF VJG RTQQH VJCV KU C
PGEGUUCT[
PCVWTCNN[ FGHKPGF YQWNF TGSWKTG C XGTUKQP QH VJG CZKQO QH EJQKEG
PCVWTCN
HQTOWNCVKQP QH PQPKPHNCVKPI
EQPFKVKQP HQT PQPKPHNCVKPI CU
O
#V VJG UCOG VKOG + YKNN CFF C IQQF FGCN QH DQQMMGGRKPI VQ (KPG U RTQQH VQ VWTP VJG OQFGN EQPUVTWEVKQP KPVQ C RTQQH QH KPVGTRTGVCDKNKV[ C VQRKE (KPG FQGU PQV GZRNKEKVN[ CFFTGUU
O
6JKU EJCTCEVGTK\CVKQP UJQWNF DG SWCNKHKGF D[ C TGEQIPKVKQP VJCV (KPG U CRRTQCEJ FQGU KPFKTGEVN[ OCMG UWTTQICVGU HQT EGTVCKP GZVGPUKQPU CXCKNCDNG CU YG JCXG UGGP 6JG
Q
R
(KPG UGV VJGQT[
VQ DG KPVTQFWEGF
UJQTVN[ KU FKTGEVN[ TGNCVGF VQ VJG IGPGTCN VJGQT[ QH CDUVTCEVKQP CU YGNN +V KU C YGNN MPQYP VJQWIJ PQV RGTJCRU XGT[ YGNN WPFGTUVQQF RJGPQOGPQP QH OCVJGOCVKECN RU[EJQNQI[ VJCV QHVGP VJG OGTG CPPQWPEGOGPV VJCV C IGPKWU JCU RTQXGF UQOG TGUWNV YKVJQWV CP[ KPFKECVKQP QH VJG RTQQH OC[ UWHHKEG C VQ GPCDNG QT KPURKTG C UWDIGPKWU VQ RTQFWEG C RTQQH QH VJG UCOG TGUWNV CPF D VQ GPCDNG QT KPURKTG C PQPIGPKWU VQ RTQFWEG C RTQQH QH C RCTVKCN TGUWNV YKVJ UVTQPIGT J[RQVJGUKU CPFQT YGCMGT EQPENWUKQP 6JG RTGUGPV OCVGTKCN TGRTGUGPVU CP KPUVCPEG QH RJGPQOGPQP D 6JG TGUWNV GZRQWPFGF KP VJKU UGEVKQP YCU CU OWEJ CU + YCU CDNG VQ YQTM QWV HQT O[UGNH QP NGCTPKPI QH C UVTQPIGT TGUWNV QH *CTXG[ (TKGFOCP YKVJQWV CEVWCNN[ UGGKPI JKU RTQQHU
M
CPF YKVJQWV CV VJG VKOG DGKPI CYCTG QH VJG GCTNKGT YQTM QH
$GTPC[U KP YJQUG HQQVUVGRU + JCF DGGP WPYKVVKPIN[ HQNNQYKPI CNN VJG YJKNG + CO ITCVGHWN VQ -CK 9GJOGKGT
O
HQT DTKPIKPI $GTPC[U RCRGT VQ O[ CVVGPVKQP VQ 5VGHCP .GWGPDGTIGT HQT JGNR YJKNG + YCU UVTWIINKPI YKVJ VJG )GTOCP CPF VQ 5QNQOQP (GHGTOCP HQT RQKPVKPI QWV VJCV VJG UVTWIING YCU WPPGEGUUCT[ UKPEG VJGTG GZKUVU CP 'PINKUJ VTCPUNCVKQP 5GG *TDCEGM CPF ,GEJ %JCRVGT 4GCNN[ KP X
4
QWIJV VQ DG TGUVTKEVGF VQ TGNCVKQPU UWEJ VJCV
4WX
KORNKGU
:W
CPF
:X
HQT CNN
W
CPF
DWV VJKU VGEJPKECN RQKPV KU KTTGNGXCPV VQ VJG OCKP EQPEGRVWCN KUUWG CPF YKNN DG KIPQTGF DGNQY 1HHKEKCNN[ YG CTG UVKNN YQTMKPI YKVJ OQPCFKE UGEQPFQTFGT NQIKE UQ VJG F[CFKE
UGEQPFQTFGT SWCPVKHKGT KU TGCNN[ CP CDDTGXKCVKQP HQT UQOGVJKPI KPXQNXKPI C OQPCFKE UGEQPFQTFGT SWCPVKHKGT CPF OGPVKQP QH QTFGTGF RCKTU DWV VJKU VGEJPKECN RQKPV KU KTTGNGXCPV VQ VJG OCKP EQPEGRVWCN KUUWG CPF YKNN DG KIPQTGF DGNQY +V OC[ DG OGPVKQPGF RCTGPVJGVKECNN[ VJCV
OCP[ DWV PQV CNN
PGY UGVVJGQTGVKE CZKQOU YQWNF DG
GSWKXCNGPV VQ PGY NQIKECN CZKQOU 6JG VTWVJ QH CP[ EQPLGEVWTG KP PWODGT VJGQT[ GZRTGUUKDNG CU C HQTOWNC QH HKTUV QT UGEQPFQTFGT 2GCPQ CTKVJOGVKE EQPUKUVGPV HQT KPUVCPEG
M
M
ψ
VJG CTKVJOGVKECN HQTOWNC EQFKPI VJG UVCVGOGPV VJCV ($ KU
KU GSWKXCNGPV VQ VJG UVCPFCTF CPF JGPEG KPVWKVKXG XCNKFKV[ QH C EGTVCKP
UGEQPFQTFGT HQTOWNC PCOGN[
φ → ψ YJGTG φ KU VJG EQPLWPEVKQP QH CNN VJG PQPNQIKECN CZKQOU QH 2 QH
YJKEJ VJGTG CTG QPN[ HKPKVGN[ OCP[ EQWPVKPI VJG EQORTGJGPUKQP CZKQOU CU NQIKECN TCVJGT VJCP PQPNQIKECN 6JG VTWVJ QH VJG EQPVKPWWO J[RQVJGUKU KU GSWKXCNGPV VQ VJG XCNKFKV[ QH CPQVJGT UGEQPFQTFGT HQTOWNC $WV VJG VTWVJ QH UQOG NCTIG ECTFKPCN J[RQVJGUGU KU PQV
4GHGTGPEGU #LFWMKGYKE\ -CUKOKGT\
Q
#PVQPGNNK #NFQ CPF /C[ 4QDGTV ! $CT*KNNGN ;GJQUJWC $GJOCPP *GKPTKEJ
Q
Q
Q
Q
O
5VWFKC 2JKNQUQRJKEC
R
(TGIG U 1VJGT 2TQITCO
1P 5[PVCEVKECN %CVGIQTKGU
R
Q
7PRWDNKUJGF /5
,QWTPCN QH 5[ODQNKE .QIKE
O
R
(TGIG U 6JGQTGO KP C %QPUVTWEVKXG 5GVVKPI
$GPCEGTTCH 2CWN
R
$GKVT¼IG \WT #NIGDTC FGT .QIKM KPUDGUQFGTG \WO 'PVUEJGKFWPIURTQDNGO
/CVJGOCVKUEJG #PPCNGP
$GNN ,QJP .
R
&KG U[PVCMVKUEJG -QPPGZKV¼V
R
9JCV 0WODGTU %QWNF 0QV $G
,QWTPCN QH 5[ODQN .QIKE
2JKNQUQRJKECN 4GXKGY
4GRTKPVGF KP
$GPCEGTTCH CPF 2WVPCO
LLLLLLLLLL Q/CVJGOCVKECN 6TWVJR ,QWTPCN QH 2JKNQUQRJ[ 4GRTKPVGF KP $GPCEGTTCH CPF 2WVPCO
LLLLLLLLLL CPF 2WVPCO *KNCT[ 2JKNQUQRJ[ QH /CVJGOCVKEU 5GNGEVGF 4GCFKPIU PF GF %CODTKFIG %CODTKFIG 7PKXGTUKV[ 2TGUU $GTPC[U 2CWN
Q<WT (TCIG FGT 7PGPFNKEJMGKVUUEJGOCVC KP FGT CZKQOCVKUEJGP /GPIGPNGJTGR KP
'UUC[U QP VJG (QWPFCVKQPU QH /CVJGOCVKEU
'F ;GJQUJWC $CT*KNNGN GV CN ,GTWUCNGO
/CIPGU 2TGUU 6TCPU , $GNN CPF / 2NCPKV\ #ZKQOCVKE 5GV 6JGQT[R +P
Q1P VJG 2TQDNGO QH 5EJGOCVC QH +PHKPKV[ KP
5GVU CPF %NCUUGU 1P VJG 9QTM D[ 2CWN $GTPC[U
'F )GTV /×NNGT
#OUVGTFCO 0QTVJ *QNNCPF $QQNQU )GQTIG 5
Q6Q $G +U VQ $G VJG 8CNWG QH C 8CTKCDNG QT 5QOG 8CNWGU QH 5QOG 8CTKCDNGU R
,QWTPCN QH 2JKNQUQRJ[
4GRTKPVGF KP $QQNQU
LLLLLLLLLL C Q0QOKPCNKUV 2NCVQPKUOR 2JKNQUQRJKECN 4GXKGY 4GRTKPVGF KP $QQNQU
LLLLLLLLLL D Q4GCFKPI VJG $GITKHHUUEJTKHVR /KPF 4GRTKPVGF KP &GOQRQWNQU 4GRTKPVGF KP $QQNQU
LLLLLLLLLL Q6JG %QPUKUVGPE[ QH (TGIGOU (QWPFCVKQPU QH #TKVJOGVKER +P 1P $GKPI CPF 5C[KPI 'UUC[U HQT 4KEJCTF %CTVYTKIJV
'F , , 6JQOUQP %CODTKFIG /+6 2TGUU 4GRTKPVGF KP
&GOQRQWNQU 4GRTKPVGF KP $QQNQU
LLLLLLLLLL Q+VGTCVKQP #ICKPR 2JKNQUQRJKECN 6QRKEU 4GRTKPVGF KP $QQNQU LLLLLLLLLL Q6JG 5VCPFCTF QH 'SWCNKV[ QH 0WODGTUR +P /GCPKPI CPF /GVJQF 'UUC[U KP *QPQT QH *KNCT[ 2WVPCO
'F )GQTIG $QQNQU %CODTKFIG %CODTKFIG 7PKXGTUKV[ 2TGUU 4GRTKPVGF
KP &GOQRQWNQU 4GRTKPVGF KP $QQNQU
LLLLLLLLLL Q9JGPEG VJG %QPVTCFKEVKQP!R #TKUVQVGNKCP 5QEKGV[ 5WRRNGOGPV 4GRTKPVGF KP $QQNQU
LLLLLLLLLL C Q!R +P 4GCFKPI 2WVPCO 'F 2GVGT %NCTM CPF $QD *CNG 1ZHQTF $NCEMYGNN 4GRTKPVGF KP $QQNQU
LLLLLLLLLL D Q6JG #FXCPVCIG QH *QPGUV 6QKN QXGT 6JGHVR +P /CVJGOCVKEU CPF /KPF 'F #NGZCPFGT )GQTIG 1ZHQTF 1ZHQTF 7PKXGTUKV[ 2TGUU 4GRTKPVGF KP $QQNQU
LLLLLLLLLL Q1P VJG 2TQQH QH (TGIGOU 6JGQTGOR +P $GPCEGTTCH CPF *KU %TKVKEU 'F #FCO /QTVQP 1ZHQTF $NCEMYGNN 4GRTKPVGF KP $QQNQU
LLLLLLLLLL .QIKE .QIKE CPF .QIKE %CODTKFIG *CTXCTF 7PKXGTUKV[ 2TGUU LLLLLLLLLL CPF $WTIGUU ,QJP 2 CPF ,GHHTG[ 4KEJCTF % %QORWVCDKNKV[ CPF .QIKE VJ GFKVKQP %CODTKFIG %CODTKFIG 7PKXGTUKV[ 2TGUU $WTCNK(QTVK %GUCTG QTKIKPCNN[ RWDNKUJGF
Q# 3WGUVKQP QH 6TCPUHKPKVG 0WODGTU 1P
9GNN1TFGTGF %NCUUGUR 6TCPU , XCP *GKLGPQQTV +P XCP *GKLGPQQTV $WTIGUU ,QJP 2 4GXKGY QH 9TKIJV
2JKNQUQRJKECN 4GXKGY
LLLLLLLLLL CPF *C\GP # 2 Q#TKVJOGVKE CPF 2TGFKECVKXG .QIKER 0QVTG &COG ,QWTPCN QH (QTOCN .QIKE
LLLLLLLLLL CPF 4QUGP )KFGQP # 5WDLGEV YKVJ 0Q 1DLGEV 5VTCVGIKGU HQT 0QOKPCNKUVKE +PVGTRTGVCVKQP QH /CVJGOCVKEU
1ZHQTF 1ZHQTF 7PKXGTUKV[ 2TGUU
%CPVQT )GQTI 4GXKGY QH (TGIG %JKJCTC %JCTNGU
1PVQNQI[ CPF VJG 8KEKQWU %KTENG 2TKPEKRNG
%QNNKPU )GQTIG ( CPF *CNRGTP , & &COG ,QWTPCN QH (QTOCN .QIKE
+VJCEC %QTPGNN 7PKXGTUKV[ 2TGUU
Q1P VJG +PVGTRTGVCDKNKV[ QH #TKVJOGVKE KP 5GV 6JGQT[R 0QVTG
,QWTPCN QH 5[ODQNKE .QIKE
&GFGMKPF 4KEJCTF
9CU UKPF WPF YCU UQNNGP FKG
9QQFTWHH $GOCP 0WODGTU
Q6JTGG 7UGU QH VJG *GTDTCPF)GPV\GP 6JGQTGO KP 4GNCVKPI /QFGN 6JGQT[ CPF 2TQQH
%TCKI 9KNNKCO 6JGQT[R
&GWVUEJG .KVGTCVWT\GKVWPI
$TWPUEJYGKI 8KGYGI 6TCPU 9QQUVGT
Q6JG 0CVWTG CPF /GCPKPI QH 0WODGTUR +P 'UUC[U QP VJG 6JGQT[ QH
%JKECIQ 1RGP %QWTV
&GOQRQWNQU 9KNNKCO GF
(TGIGOU 2JKNQUQRJ[ QH /CVJGOCVKEU
%CODTKFIG *CTXCTF 7PKXGTUKV[
2TGUU &WOOGVV /KEJCGN
(TGIG 2JKNQUQRJ[ QH /CVJGOCVKEU
(GHGTOCP 5QNQOQP CPF *GNNOCP )GQHHTG[ 2JKNQUQRJKECN .QIKE
(GTTGKTC (GTPCPFQ !
%CODTKFIG *CTXCTF 7PKXGTUKV[ 2TGUU
Q2TGFKECVKXG (QWPFCVKQPU QH #TKVJOGVKER ,QWTPCN QH
Q#OGPFKPI (TGIGOU )TWPFIGUGV\G FGT /CVJGOCVKMR 7PRWDNKUJGF /5
LLLLLLLLLL CPF 9GJOGKGT -CK Q1P VJG %QPUKUVGPE[ QH VJG ∆ %# (TCIOGPV QH (TGIGOU R ,QWTPCN QH 2JKNQUQRJKECN .QIKE
)TWPFIGUV\G
(KPG -KV
6JG .KOKVU QH #DUVTCEVKQP
(TGIG )QVVNQD &GPMGPU
1ZHQTF 1ZHQTF 7PKXGTUKV[ 2TGUU
$GITKHHUUEJTKHV GKPG FGT CTKVJOGVKUEJGP PCEJIGDKNFGVG (QTOGNURTCEJG FGU TGKPGP
*CNNG .QWKU 0GDGTV 6TCPU 5VGHCP $CWGT/GPIGNDGTI +P XCP *GKLGPQQTV
LLLLLLLLLL &KG )TWPFNCIGP FGT #TKVJOGVKM 'KPG NQIKUEJOCVJGOCVKUEJG 7PVGTUWEJWPI WDGT FGP $GITKHH FGT
$TGUNCW 9KNJGNO -QGDPGT 6TCPU , . #WUVKP
(QWPFCVKQPU QH #TKVJOGVKE
1ZHQTF $NCEMYGNN
LLLLLLLLLL Q¯DGT 5KPP WPF $GFGWVWPIR
Q1P 5GPUG CPF 4GHGTGPEGR +P )GCEJ CPF $NCEM
LLLLLLLLLL )TWPFIGUGV\G FGT #TKVJOGVKM DGITKHHUUEJTKHVNKEJ CDIGNGKVGV XQNU ,GPC 2QJNG 4GRTKPVGF *KNFGUJGKO 1NOU 2CTVKCN VTCPU 2'$ ,QWTFCKP CPF , 5VCEJGNTQVJ QH #TKVJOGVKE
$CUKE .CYU
+P )GCEJ CPF $NCEM
(TKGFOCP *CTXG[ !
Q5GPVGPVKCN 4GHNGEVKQP CPF 'NGOGPVCN 5GPVGPVKCN 4GHNGEVKQPR 7PRWDNKUJGF /5
)GCEJ 2GVGT 4GXKGY QH / &WOOGVV
(TGIG 2JKNQUQRJ[ QH .CPIWCIG /KPF
LLLLLLLLLL CPF $NCEM /CZ GFU 6TCPUNCVKQPU HTQO VJG 9TKVKPIU QH )QVVNQD (TGIG 1ZHQTF $NCEMYGNN )QNFHCTD 9CTTGP
Q(KTUV1TFGT (TGCIGCP 6JGQT[ KU 7PFGEKFCDNGR ,QWTPCN QH 2JKNQUQRJKECN .QIKE
*CNG $QD
Q4GCNU D[ #DUVTCEVKQPR 2JKNQUQRJKC /CVJGOCVKEC
*¶LGM 2GVT CPF 2WFNCM 2CXGN *CNNGVV /KEJCGN %CPVQTKCP *CTTKPIVQP .GQ GV CN GFU
/GVCOCVJGOCVKEU QH (KTUV1TFGT #TKVJOGVKE
5GV 6JGQT[ CPF .KOKVCVKQP QH 5K\G
$GTNKP 5RTKPIGT
1ZHQTF 1ZHQTF 7PKXGTUKV[ 2TGUU
*CTXG[ (TKGFOCPOU 4GUGCTEJ QP VJG (QWPFCVKQPU QH /CVJGOCVKEU
#OUVGTFCO 0QTVJ *QNNCPF *C\GP # 2 4GXKGY QH 9TKIJV *GEM ,T 4KEJCTF )
#WUVTCNCUKCP ,QWTPCN QH 2JKNQUQRJ[
Q1P VJG %QPUKUVGPE[ QH 5GEQPF1TFGT %QPVGZVWCN &GHKPKVKQPUR 0QÖU
LLLLLLLLLL Q6JG &GXGNQROGPV QH #TKVJOGVKE KP (TGIGOU )TWPFIGUGV\G FGT #TKVJOGVKMR ,QWTPCN QH 5[ODQNKE .QIKE
4GRTKPVGF YKVJ C RQUVUETKRV KP &GOQRQWNQU
LLLLLLLLLL Q1P VJG %QPUKUVGPE[ QH 2TGFKECVKXG (TCIOGPVU QH (TGIGOU )TWPFIGUGV\G FGT #TKVJOGVKM
R *KUVQT[ CPF 2JKNQUQRJ[ QH .QIKE
LLLLLLLLLL C Q6JG (KPKVG CPF VJG +PHKPKVG KP (TGIGR
+P
2JKNQUQRJ[ QH /CVJGOCVKEU 6QFC[
'F / 5EJKTP 1ZHQTF 1ZHQTF 7PKXGTUKV[ 2TGUU
LLLLLLLLLL D Q(KPKVWFG CPF *WOGOU 2TKPEKRNGR ,QWTPCN QH 2JKNQUQRJKECN .QIKE *QFGU *CTQNF
Q.QIKEKUO CPF VJG 1PVQNQIKECN %QOOKVOGPVU QH #TKVJOGVKER ,QWTPCN QH 2JKNQUQRJ[
*TDCEGM -CTGN CPF ,GEJ 6JQOCU -CPCOQTK #MKJKTQ .KPPGDQ «[UVGKP !
+PVTQFWEVKQP VQ 5GV 6JGQT[
6JG *KIJGT +PHKPKVG
TF GF 0GY ;QTM /CTEGN &GMMGT
EQTTGEVGF RTKPVKPI $GTNKP 5RTKPIGT
Q2TGFKECVKXG (TCIOGPVU QH (TGIG #TKVJOGVKER 7PRWDNKUJGF /5
.ÑYGPJGKO .GQRQNF
Q¯DGT /ÑINKEJMGKVGP KO 4GMCVKXMCNM×NR /CVJGOCVKUEJG #PPCNGP
6TCPU 5VGHCP $CWGT/GPIGNDGTI
Q1P 2QUUKDKNKVKGU KP VJG %CNEWNWU QH 4GNCVKXGUR +P XCP
*GKLGPQQTV /QPVCIPC (TCPEQ CPF /CPEKPK #PVQPGNNC ,QWTPCN QH (QTOCN .QIKE
/QUVQYUMK #PFT\GL /CVJGOCVKE¿
Q# /KPKOCN 2TGFKECVKXG 5GV 6JGQT[R 0QVTG &COG
Q5QOG +ORTGFKECVKXG &GHKPKVKQPU KP VJG #ZKQOCVKE 5GV 6JGQT[R (WPFCOGPVC
0GNUQP 'FYCTF
2TGFKECVKXG #TKVJOGVKE
2TKPEGVQP 2TKPEGVQP 7PKXGTUKV[ 2TGUU
0QXCM )CN +NUG
Q# %QPUVTWEVKQP QH /QFGNU QH %QPUKUVGPV 5[UVGOUR (WPFCOGPVC /CVJGOCVKE¿
2CTUQPU %JCTNGU
Q(TGIGOU 6JGQT[ QH 0WODGTR +P 2JKNQUQRJ[ KP #OGTKEC 'F /CZ $NCEM
+VJCEC %QTPGNN 7PKXGTUKV[ 2TGUU 4GRTKPVGF KP &GOQRQWNQU 2CTUQPU 6GTGPEG
Q1P VJG %QPUKUVGPE[ QH VJG (KTUV1TFGT 2QTVKQP QH (TGIGOU .QIKECN 5[UVGOR 0QVTG
&COG ,QWTPCN QH (QTOCN .QIKE
3WKPG 9 8
4GRTKPVGF KP &GOQRQWNQU
Q1P (TGIGOU 9C[ 1WVR /KPF 4GRTKPVGF KP 5GNGEVGF .QIKE 2CRGTU
0GY ;QTM 4CPFQO *QWUG
LLLLLLLLLL /CVJGOCVKECN .QIKE TGXKUGF GF %CODTKFIG *CTXCTF 7PKXGTUKV[ 2TGUU LLLLLLLLLL Q# $CUKU HQT 0WODGT 6JGQT[ KP (KPKVG %NCUUGUR $WNNGVKP QH VJG #OGTKECP /CVJGOCVKECN 5QEKGV[
4COUG[ ( 2
Q6JG (QWPFCVKQPU QH /CVJGOCVKEUR 2TQEGGFKPIU QH VJG .QPFQP /CVJGOCVKECN 5QEKGV[
4GRTKPVGF KP
6JG (QWPFCVKQPU QH /CVJGOCVKEU
'F 4 $ $TCKVJYCKVG
.QPFQP 2CWN 6TGPEJ 6WDPGT 4QUUGT , $CTMNG[
Q6JG $WTCNK(QTVK 2CTCFQZR ,QWTPCN QH 5[ODQNKE .QIKE
LLLLLLLLLL CPF 9CPI *CQ Q0QP5VCPFCTF /QFGNU HQT (QTOCN .QIKEUR ,QWTPCN QH 5[ODQNKE .QIKE 4WUUGNN $GTVTCPF .GVVGT VQ (TGIG 6TCPU $GXGTN[ 9QQFYCTF +P XCP *GKLGPQQTV
LLLLLLLLLL 6JG 2TKPEKRNGU QH /CVJGOCVKEU + %CODTKFIG %CODTKFIG 7PKXGTUKV[ 2TGUU LLLLLLLLLL Q5QOG &KHHKEWNVKGU KP VJG 6JGQT[ QH 6TCPUHKPKVG 0WODGTU CPF 1TFGT 6[RGUR 2TQEGGFKPIU QH VJG .QPFQP /CVJGOCVKECN 5QEKGV[
LLLLLLLLLL Q/CVJGOCVKECN .QIKE CU $CUGF QP VJG 6JGQT[ QH 6[RGUR #OGTKECP ,QWTPCN QH /CVJGOCVKEU
4GRTKPVGF KP XCP *GKLGPQQTV
5EJTQGFGT*GKUVGT 2GVGT
Q# /QFGN6JGQTGVKE 4GEQPUVTWEVKQP QH (TGIGOU 2GTOWVCVKQP #TIWOGPVR
0QVTG &COG ,QWTPCN QH (QTOCN .QIKE
5JQGPHKGNF , 4
Q# 4GNCVKXG %QPUKUVGPE[ 2TQQHR ,QWTPCN QH 5[ODQNKE .QIKE
5KORUQP 5VGRJGP ) 5VGKPGT /CTM
5WDU[UVGOU QH 5GEQPF 1TFGT #TKVJOGVKE
/CVJGOCVKECN -PQYNGFIG
$GTNKP 5RTKPIGT
+VJCEC %QTPGNN 7PKXGTUKV[ 2TGUU
6CTUMK #NHTGF CPF /QUVQYUMK #PFT\GL CPF 4QDKPUQP 4CRJCGN /
7PFGEKFCDNG 6JGQTKGU
#OUVGTFCO 0QTVJ *QNNCPF 6GPPCPV 0GKN
#PVK4GCNKUO KP .QIKE 6TWVJ CU 'VGTPCN
7TSWJCTV #NCUFCKT 'FKVQTKCN KPVTQFWEVKQPU VQ
1ZHQTF 1ZHQTF 7PKXGTUKV[ 2TGUU
%QNNGEVGF 2CRGTU QH $GTVTCPF 4WUUGNN
(QWPFCVKQPU
QH .QIKE .QPFQP 4QWVNGFIG 8CP *GKLGPQQTV ,GCP GF (TQO (TGIG VQ )ÑFGN # 5QWTEGDQQM KP /CVJGOCVKECN .QIKE %CODTKFIG *CTXCTF 7PKXGTUKV[ 2TGUU 9CPI *CQ
Q6TWVJ &GHKPKVKQPU CPF %QPUKUVGPE[ 2TQQHUR 6TCPUCEVKQPU QH VJG #OGTKECP
/CVJGOCVKECN 5QEKGV[ 9GJOGKGT -CK
Q%QPUKUVGPV (TCIOGPVU QH VJG )TWPFIGUGV\G CPF VJG 'ZKUVGPEG QH 0QP.QIKECN
1DLGEVUR 5[PVJGUG 9JKVGJGCF # 0 CPF $ 4WUUGNN 2TKPEKRKC /CVJGOCVKEC XQNU %CODTKFIG %CODTKFIG 7PKXGTUKV[ 2TGUU PF GF 9TKIJV %TKURKP (TGIGOU %QPEGRVKQP QH 0WODGTU CU 1DLGEVU #DGTFGGP 5EQVU 2JKNQUQRJKECN /QPQITCRJU
LLLLLLLLLL CPF *CNG $QD 6JG 4GCUQPOU 2TQRGT 5VWF[ 'UUC[U VQYCTFU C 0GQ(TGIGCP 2JKNQUQRJ[ QH /CVJGOCVKEU 1ZHQTF 1ZHQTF 7PKXGTUKV[ 2TGUU
+PFGZ )GPGTCNN[ QPN[ VJG HKTUV QEEWTTGPEGU QH VGEJPKECN VGTOU QT VJG HKTUV KP GCEJ UGPUG YJGTG VJG VGTO JCU UGXGTCN UGPUGU CTG KPFGZGF DGNQY +VGOU KPFGZGF KP VJG
$GTPC[U UGV VJGQT[ $ (TGIGCPK\GF
CDDTGXKCVKQPU UVCVWU QH
CDUQNWVGPGUU QH DQWPFGF HQTOWNCU CDUVTCEVKQP IGPGTCN VJGQT[ QH RTKPEKRNGU YGCM VJGQT[ CDUVTCEVU HQT CP GSWKXCNGPEG PQVCVKQP "
CFFKVKQP FGHKPKVKQP QH CUUQEKCVKXG CPF EQOOWVCVKXG NCYU QH CFFKVKXG EQPEGRV HQTOWNC MGTPGN
CNGRJQPG
DKEQPFKVKQPCN
↔
VJGQT[ $QQNQU $WTIGUU CPF ,GHHTG[ P P P P P P P
QDLGEV
VJGQTGO
$QQNQU UGV VJGQT[ $QQNQUKHKGF $GTPC[U UGV
#EMGTOCPP 9KNJGNO
ℵ
$GTPC[U 2CWN P JKU
$QODGNNK 4CHCGN
CPF h HQT
CNGRJ\GTQ
6#$.'5 CTG PQV KPENWFGF
ℵ
$QQNQU )GQTIG
CPCN[UKU 5GG 2GCPQ CTKVJOGVKE UGEQPFQTFGT
P P P
CPCN[VKECN JKGTCTEJ[
P P JKU EQPXGTUG VQ (TGIGOU
CPEGUVTCN
VJGQTGO JKU
#PVQPGNNK #NFQ P
CTKVJOGVKECN JKGTCTEJ[
DQQVUVTCRRKPI
CUUQEKCVKXG NCY HQT CFFKVKQP CPF OWNVKRNKECVKQP
$QWEJGT #PFTGY
∆
QNKVVNGR VJGQTGO
HCKNU HQT GZRQPGPVKCVKQP
DQWPFGF HQTOWNC 5GG
HQT KPVGTUGEVKQP CPF WPKQP
DQWPFGF SWCPVKHKGTU KP CTKVJOGVKE KP UGV
CZKQO UEJGOG XU UKPING CZKQO QH KPFWEVKQP QH UGRCTCVKQP CZKQOU QH EJQKEG KPHKPKV[ TGFWEKDKNKV[ 5GG WPFGT EJQKEG KPHKPKV[ TGFWEKDKNKV[
HQTOWNC
VJGQT[ DTCEMGV PQVCVKQPU n ^^ ^^ ] _ CPF = ?
〈 〉 5GG
WPFGT EQPEGRV PWODGT UGV UGIOGPV $WTCNK(QTVK %GUCTG JKU RCTCFQZ
DCFEQORCP[ QDLGEVKQP
$WTIGUU ,QJP 2
$GITKHHUUEJTKHV $GJOCPP *GKPTKEJ
%CPVQT )GQTI JKU
$GNN , .
VJGQTGO CPF RCTCFQZ JKU VTCPUHKPKVG
$GPCEGTTCH 2CWN KZ
ECTFKPCNU CPF QTFKPCNU
%CTFCPQ )KTQNCOQ P
EQPUVTWEVKDKNKV[ CZKQO QH 8 .
ECTFKPCNU UGG VTCPUHKPKVG
EQPVKPWWO
%CTVGUKCP RTQFWEV
⊗
c
EQPVKPWWO J[RQVJGUKU %*
ECVGIQTKGU 0 5 CPF FGTKXCVKXGU P
%TCKI KPVGTRQNCVKQP VJGQTGO
%JKJCTC %JCTNGU P P
%TCKI 9KNNKCO P
EJQKEG CZKQO QH #%
EWVGNKOKPCVKQP VJGQTGO P
P %JWTEJOU VJGQTGO VJGUKU
&GFGMKPF HKPKVWFG
ENCUUGU
&GFGMKPFOU VJGQTGO XGTUKQP ++
EQCPCN[VKE UGVU
P XGTUKQP +++
EQHKPKVG EQUOCNN
&GFGMKPF 4KEJCTF P
EQFG PWODGT
FGFWEKDKNKV[ XU EQPUGSWGPEG
EQGZVGPUKXGPGUU
≡
FGHKPGF XU RTKOKVKXG PQVCVKQP
%QJGP 2CWN
FGHKPKVKQP KORNKEKV D[ TGEWTUKQP
%QNNKPU )GQTIG
&GNVCHQTOWNCU
EQOOWVCVKXG NCY HQT CFFKVKQP CPF
∆
QT
VJGQT[
CPF WPKQP
∆
C
EQORNGZ XU UKORNG 5GG \KI\CI VJGQT[ EQORQUKVKQP QH HWPEVKQPU EQORTGJGPUKQP HQT UGVU KP CPCN[UKU EQORWVCVKQP QH RTQFWEV QH RQYGT QH UWO EQPEGRV PQVCVKQP n EQPFKVKQPCN
→
EQPLWPEVKQP EQPUGSWGPEG XU FGFWEKDKNKV[
QT DQWPFGF KP
CTKVJOGVKE
OWNVKRNKECVKQP HQT KPVGTUGEVKQP
EQORNGZ PWODGTU
∆
∆
∆
QT DQWPFGF KP UGV
QT CTKVJOGVKECN
EQORTGJGPUKQP P
FGOQPUVTCDKNKV[ XU XCNKFKV[ FGUETKRVKQPU 4WUUGNNOU VJGQT[ CPF PQVCVKQP ι P FKTGEVKQPU (TGIGOU GZCORNG FKULWPEVKQP
∨
FKUVKPEVPGUU
l
FKUVTKDWVKXG NCY FQWDNG RTGFKECVKXG 5GG WPFGT RTGFKECVKXG &WOOGVV /KEJCGN F[CFKE JKIJGTQTFGT NQIKE XU OQPCFKE
EQPUGTXCVKXG GZVGPUKQP
∈
EQPUKUVGPE[ XU UCVKUHKCDKNKV[
GNGOGPVJQQF
EQPUKUVGPE[ RTQQHU QH OQFGNVJGQTGVKE
GRKUVGOQNQIKECN ENCKOU
RTQQHU QH RTQQHVJGQTGVKE
GRUKNQP ε U[ODQN *KNDGTVOU
TGNCVKXG CPF EQPUKUVGPE[ UVTGPIVJ
GRUKNQP ∈ OQFGNU 5GG OQFGNU
EQPUKUVGPE[ VJGQTGOU
GSWKPWOGTCVKQP
EQPUVCPVU KPVTQFWEVKQP QH
GSWKPWOGTQUKV[
≈
GSWKXCNGPEG TGNCVKQP OWNVKRNG UVTQPIN[ UWKVCDNG UWKVCDNG YGCMN[
HKTUVQTFGT 2GCPQ CTKVJOGVKE 5GG 2GCPQ CTKVJOGVKE HKTUVQTFGT HKTUVTQWPF 5GG WPFGT RTGFKECVKXG
UWKVCDNG GSWKXCNGPEG ENCUU 5GG UGV QH GSWKXCNGPVU
(KVEJ (TGFGTKE
GZEGNNGPV
HNQCVKPI EQPEGRVU
GZENWFGF OKFFNG
HQWPFCVKQP CZKQO
GZGIGVKECN ENCKOU
(TCPMGN #DTCJCO 5GG TGRNCEGOGPV CZKQO
GZKUVGPVKCN SWCPVKHKGT
∃ 5GG WPFGT SWCPVKHKGT
GZRQPGPVKCN EQPEGRV HQTOWNC MGTPGN GZRQPGPVKCVKQP
↑ FGHKPKVKQP QH KP
U[UVGOU QH CTKVJOGVKE
(TGIG CTKVJOGVKE (#
Π
XGTUKQP
(TGIGOU VJGQTGO (KPGOU TGHKPGOGPV CPF EQPXGTUG )GCEJ$QQNQU
GZVGPUKQPCN JKIJGTQTFGT EQPEGRV
EQPXGTUG *GEM$GNN TGHKPGOGPV
GZVGPUKQPCNKV[ HQT EQPEGRVU
.KPPGDQ TGHKPGOGPV
HQT UGVU GZVGPUKQPU PQVCVKQPU e CPF
RTGFKECVKXG
U CPF ] _ HQT
UCHG
(TGIGOU YC[ QWV (TGIG )QVVNQD
HCEVQTKCN
P P P
HCKT IQQF XGT[ IQQF GZEGNNGPV
P P P JKU
HCNNKPI WPFGT
∇
(GHGTOCP 5QNQOQP KZ P P (GTTGKTC (GTPCPFQ P (KPG UGV VJGQT[ P (KPG -KV KZ JKU IGPGTCN VJGQT[ QH CDUVTCEVKQP JKU
FGHKPKVKQPU CPF EJCKP QH FGFWEVKQPU (TGIGCPK\GF $GTPC[U UGV VJGQT[ ($ 5GG WPFGT $GTPC[U UGV VJGQT[ (TKGFOCP *CTXG[ KZ P P
QNKVVNGR VJGQTGO JKU TGHKPGOGPV QH
HWPEVKQPU[ODQN KPVTQFWEVKQP QH
(TGIGOU VJGQTGO CPF EQPXGTUG
HWPEVKQPU KP (TGIGOU UGPUG
P P HKPKVGN[OCP[
# 5GG SWCPVKHKGT
HKPKVKUO P HKPKVKUVKE XU KPHKPKVKUVKE HKPKVWFG 5GG &GFGMKPF HKPKVWFG OKPKOCNN[ CFGSWCVG HKPKVWFG 4WUUGNN HKPKVWFG HKTUVQTFGT NQIKE NCPIWCIGU CPF VJGQTKGU
)CWUU % ( )GCEJ 2GVGT P IGPGTCNK\GF EQPVKPWWO J[RQVJGUKU )%* IGPGTCNK\GF SWCPVKHKGTU 5GG SWCPVKHKGTU )GPV\GP CTKVJOGVKE +∆ UWRGTGZR )GPV\GP )GTJCTF INQDCN KPVGTRTGVCDKNKV[ INQDCN XU NQECN CZKQO QH EJQKEG
)ÑFGN EQORNGVGPGUU VJGQTGO
J[RGTKPCEEGUUKDNG ECTFKPCNU
)ÑFGN KPEQORNGVGPGUU VJGQTGOU )ÑFGN -WTV
KFGPVKV[ HWPEVKQP
)QNFHCTD 9CTTGP P
KFGPVKV[ RTQDNGO
IQQF XGT[ IQQF
KFGPVKV[ KP OCP[UQTVGF CPF JKIJGTQTFGT NQIKE P
ITCOOCT (TGIGCP CPF UGEQPFQTFGT SWCPVKHKECVKQP
KORNKEKV FGHKPKVKQP
)TGNNKPI RCTCFQZ 5GG JGVGTQNQIKECN RCTCFQZ
KORTGFKECVKXG XU RTGFKECVKXG 5GG RTGFKECVKXKV[
)TWPFIGUGV\G
KPCEEGUUKDNG ECTFKPCNU
P
KPEQORNGVGPGUU QH UGEQPFQTFGT NQIKE
)TWPFNCIGP P
5GG CNUQ )ÑFGN
)T\GIQTE\[M #PFT\GL
KPEQORNGVGPGUU VJGQTGOU KPFGHKPKVGPGUU CPF KPFGHKPCDKNKV[
*CLGM 2GVT
KPFGUETKDCDNG ECTFKPCNU
*CNG $QD
KPFKUEGTPKDKNKV[
*CNNGVV /KEJCGN P
KPFKXKFWCNU
*CNRGTP , &
KPFWEVKDNG
*COKNVQP 9KNNKCO P
KPFWEVKQP OCVJGOCVKECN CZKQO +
∆
*CTTKPIVQP .GQ P
XU UEJGOG +
*C\GP # 2 KZ P
SWCPVKHKGTHTGG
*GEM 4KEJCTF KZ
Σ Σ
KPHKPKVKUVKE XU HKPKVKUVKE
i P vi P
*GNNOCP )GQHHTG[ P
KPHKPKV[
*GPMKP OQFGNU 5GG WPFGT OQFGNU
KPHKPKV[ CZKQO QH
JGVGTQNQIKECN RCTCFQZ
KPHNCVKQP CPF J[RGTKPHNCVKQP P
JKIJGT RTGFKECVGU CPF EQPEGRVU JKIJGTQTFGT
KPHQTOCVKQP RTQDNGO
NQIKE *KNDGTV
εU[ODQN
KPLGEVKQP KPUVCPEG QH C UEJGOG
Z
*KNDGTVOU RTQITCO
KPVGIGTU
*KNDGTV &CXKF
KPVGTRQNCPV
*QFGU *CTQNF
KPVGTRTGVCVKQP QH 3
JQTUG EQPEGRV QH
KPVWKVKQP OCVJGOCVKECN
*TDCEGM -CTGN P
KPVWKVKQPKUO P
*WOGOU RTKPEKRNG *2 CPF RTGFKECVKXKV[
KPFWEVKXG EQPEGRV HQTOWNC UGV
P JKU EQPUKUVGPE[ VJGQTGO P
5GG
%TCKI KPVGTRQNCVKQP VJGQTGO
KPXCTKCPEG FQWDNG PGCT CPF XGT[ PGCT YKVJ TGURGEV VQ
.ÑYGPJGKO5MQNGO VJGQTGO
KPXGTUG QH C HWPEVKQP 5GG
.ÑYGPJGKO$GJOCPP VJGQTGO P P P
C UGV
KQVC PQVCVKQP
.ÑYGPJGKO .GQRQNF
FGUETKRVKQPU
KTTGHNGZKXKV[
/CFF[ 2GPGNQRG KZ
KUQOQTRJKUO
/CPEKPK #PVQPGNNC OCP[UQTVGF NCPIWCIGU CPF NQIKEU OQFGNU HQT
,GEJ 6JQOCU P
`
,GHHTG[ 4KEJCTF
/CVJGOCVKEC
,WNKWU %CGUCT RTQDNGO
OCZKOCNKUV CPF PQP XGTUKQP QH NKOKVCVKQP QH
P
UK\G -CNOCT CTKVJOGVKE +∆ GZR
/C[ 4QDGTV P
-CNOCT .CU\NQ
/KNN ,QJP 5VWCTV
-CPCOQTK #MKJKTQ KZ
OKPKOCNN[ CFGSWCVG HKPKVWFG
-CPV +OOCPWGN
/KPVU )TKIQTK P
-GKUNGT * , P
/. /CVJGOCVKECN
-NGGPG JKGTCTEJ[
OQFGN VJGQT[
-TKRMG 5CWN KZ
OQFGNU HQT OCP[UQTVGF HKTUVQTFGT CPF HQT
XU
.QIKE
RTQQH VJGQT[
UGEQPFQTFGT VJGQTKGU EQPUKUVGPE[ RTQQHU WUKPI
NCTIG ECTFKPCNU .CY 8 PWOGTKECN
08
∈OQFGNU *GPMKP OQFGNU
.GKDPK\ ) 9
UVCPFCTF OQFGNU 6CTUMK
.GUPKGYUMK 5VCPKUNCY P
OQFGNU
.GWGPDGTIGT 5VGHCP KZ
OQPCFKE HKTUVQTFGT NQIKE YKVJ KFGPVKV[
.GX[ JKGTCTEJ[
JKIJGTQTFGT NQIKE
.GX[ #\TKGN
XU
NKOKVCVKQP QH UK\G
/QPVCIPC (TCPEQ
NKPGCT QTFGT
/QTUG-GNN[ UGV VJGQT[ /-
.KPPGDQ «[UVGKP KZ JKU
/QUVQYUMK #PFT\GL P
TGHKPGOGPV QH (TGIGOU VJGQTGO NQECN KPVGTRTGVCDKNKV[ NQECN
XU
NQIKE
INQDCN CZKQO QH EJQKEG
5GG
HKTUVQTFGT UGEQPFQTFGT JKIJGTQTFGT
NQIKECN QDLGEVU NQIKECN PQVKQPU CPF KPXCTKCPEG
OWNVKRNG GSWKXCNGPEG
5GG
F[CFKE
GSWKXCNGPEG
OWNVKRNKECVKQP ⋅ FGHKPKVKQP QH CUUQEKCVKXG CPF EQOOWVCVKXG NCYU QH OWNVKRNKECVKXG EQPEGRV HQTOWNC MGTPGN QDLGEV
PCOGU
5GG
2GCPQ CTKVJOGVKEU HKTUVQTFGT
ECVGIQT[ 0
PCVWTCN ECPFKFCVG OQFGN QH UGV VJGQT[ PCVWTCN PWODGTU
Π
XGTUKQP UGEQPFQTFGT
N FGHKPKVKQPU QH
VJKTFQTFGT
PGICVKQP `
2GCPQ RQUVWNCVGU 22
0GNUQP 'FYCTF KZ
RGTOWVCVKQP
0GNUQP9KNMKG CTKVJOGVKE +∆
2KHQTOWNCU
Π Π KP CTKVJOGVKE Π Π KP
UGV VJGQT[
PGQ(TGIGCPKUO
Π
P P
Π Π
CPF
Π
KP CPCN[UKU PQVCVKQP HQT
Π EQORTGJGPUKQP
0( Q0GY (QWPFCVKQPUR
RTQFWEV
PQ ENCUUGU VJGQT[
Π 2GCPQ CPF (TGIG CTKVJOGVKEU
PQOKPCNKUO
0QXCE )CN +NUG P
RNWTCN NQIKE
PWODGTU PQVCVKQP HQT CPF ^^ ^^ HQT
2QKPECTÃ *GPTK P P
PWOGTKECN GSWKXCNGPEG CPF .CY 8 PWOGTKECN SWCPVKHKGTU
5GG
.CY 8
∃M 5GG SWCPVKHKGT
RQN[CFKE JKIJGTQTFGT NQIKE RQYGT UGV CZKQO NKOKVGF HQTOU
℘P ω
UGV
VJGQT[ YKVJQWV KV <( QDLGEV P CFFKVKQPCN QDLGEV QOGIC
ο
ω QOGICQPG ω
QPGVQQPG EQTTGURQPFGPEG QTFGT QT
QP PCVWTCN PWODGTU FGHKPKVKQPU QH
RTGUGTXCVKQP QH
RQYGTHWNPGUU RTCIOCVKUO RTGFKECVGU RTGFKECVKXKUO P RTGFKECVKXKV[ HKTUV CPF UGEQPFTQWPF CPF *WOGOU
V[RGU CPF QTFKPCN PWODGTU QTFGTGF RCKT
〈 〉 9GKPGT-WTCVQYUMK FGHKPKVKQP
RTKPEKRNG UKORNG FQWDNG TCOKHKGF
QH
UGEQPF FKOGPUKQP QH
QTFGTN[ EQPEGRV HQTOWNC MGTPGN
KORTGFKECVKXKV[ WNVTC
QDLGEV QTFKPCNU
5GG
VTCPUHKPKVG
RTGUGTXCVKQP QH QTFGT
5GG WPFGT
QTFGT
RTGUWRRQUKVKQP RCKT
5GG
RCTCFQZ
QTFGTGF RCKT 5GG
RTKOKVKXG TGEWTUKXG CTKVJOGVKE 24# P P
$WTCNK(QTVK RCTCFQZ JGVGTQNQIKECN
RCTCFQZ 4WUUGNNOU RCTCFQZ
RTKOKVKXG TGEWTUKXG HWPEVKQPU
RCTCOGVGTU
RTKOKVKXG
2CTUQPU CTKVJOGVKE +Σ
RTQFWEV
2CTUQPU %JCTNGU KZ
RTQFWEVKXKV[
2CTUQPU 6GTGPEG
RTQHKNG
5GG
5GG
RTQLGEVKXG
XU
FGHKPGF
XU
RTKOKVKXG
OWNVKRNKECVKQP XU
OWNVKRNKECVKXKV[
CHHKPG IGQOGVT[
RTQQH VJGQT[
XU
TGIWNCTKV[ CZKQO
OQFGN VJGQT[
5GG
HQWPFCVKQP CZKQO
RTQRQUKVKQPCN HWPEVKQPU
4GKPJCTFV ECTFKPCN P
RTQVQPCVWTCN
TGNCVKQPCN EQPEGRVU
2WFNCM 2CXGN
TGNCVKXG EQPUKUVGPE[
RWTG UGVU
TGNCVKXK\CVKQP
2[VJCIQTGCPU
TGRNCEGOGPV CZKQO
5GG
EQPEGRVU
5GG
EQPUKUVGPE[ UVTGPIVJ
TGRTGUGPVCDKNKV[ 3
5GG
TGRTGUGPVCVKXGU
4QDKPUQP CTKVJOGVKE
SWCPVKHKGTU DQWPFGF KP CTKVJOGVKE DQWPFGF KP UGV VJGQT[ GZKUVGPVKCN HKPKVGN[OCP[
∃
# IGPGTCNK\GF
PWOGTKECN
∃M
CPF
TGXGTUG OCVJGOCVKEU 4QDKPUQP CTKVJOGVKE 3 4QDKPUQP 4CRJCGN
∃M RNWTCN
5GG CNUQ
CTKVJOGVKE
TGNCVKXK\GF
4QUGP )KFGQP
UGEQPFQTFGT KPVGTRTGVCVKQP QH
4QUUGT , $ P
WPKSWG GZKUVGPEG
∃
4WUUGNN EQPEGRV HKPKVWFG P RCTCFQZ UGV
WPKXGTUCN SWCPVKVCVKXG CPF PQP XGTUKQPU QH NKOKVCVKQP QH
4WUUGNN $GTVTCPF P P P
UK\G SWCVGTPKQPU
H P
3WKPG 9 8 P P P
UCHG GZVGPUKQPU UCVKUHKCDKNKV[ KPVWKVKXG
TCOKHKGF
5GG WPFGT
UCVWTCVGF
TCPM QH UGV VGTOU ] _
UEJGOG
TCVKQPCN PWODGTU TGCN PWODGTU
Q
R GZVGPFGF P
5GG
XU
5GG
CZKQO UEJGOG
9TKIJV %TKURKP
UGEQPFQTFGT NQIKE OQPCFKE
XU
CPF
*CNG
5GG CNUQ
F[CFKE
UGEQPFQTFGT 2GCPQ CTKVJOGVKE
5GG
2GCPQ
CTKVJOGVKE UGEQPFQTFGT
TGFWEKDKNKV[ CZKQO QH
UGEQPFTQWPF
TGHGTGPEG
5GG
UGPUG
5GG
$QD
TGFWEKDKNKV[ QH QPG VJGQT[ VQ CPQVJGT
XU
XU
WPUCVWTCVGF
5EQVVKUJ UEJQQN
TGEWTUKXG UGVU TGEWTUKXGN[ GPWOGTCDNG UGVU
UVCPFCTF
5EJTQGFGT*GKUVGT 2GVGT
FGHKPKVKQP D[ TGEWTUKQP
TGEWTUKXG UCVWTCVKQP P
XU
EQPUKUVGPE[
RTGFKECVKXG
4COUG[ ( 2
TGEWTUKQP
4QDKPUQP
5GG WPFGT
RTGFKECVKXG
2GCPQ CTKVJOGVKE UGEQPFQTFGT
TGHNGEVKQP RTKPEKRNG
UGIOGPV = ?
TGHNGZKXKV[ CPF TGHNGZKXG ENQUWTG
UGPUG
TGITGUUKXG
UGPVGPEGU
5GG
TGHGTGPEG 5GG
ECVGIQTKGU 0 5 CPF FGTKXCVKXGU
UGRCTCVKQP CZKQO CZKQO
UWRGTGZRQPGPVKCVKQP
XU
⇑ FGHKPKVKQP QH KP
U[UVGOU QH CTKVJOGVKE
UEJGOG UGRCTCVKQP RTKPEKRNGU
UWRGTVTCPUKVKXG UGV
UGV QH GSWKXCNGPVU = ?
UWRGTVTCPUKVKXG UGV
UGV VJGQT[ CZKQOCVKE
U[OOGVT[ CPF U[OOGVTKE ENQUWTG
UGVJQQF PQVCVKQP HQT ²
U[PVCEVKE
UGVU ] _ PQVCVKQP HQT VGTOU
Σ
VJGQT[
Σ Σ
CPF
Σ
6CMGWVK )CKUK P
KP CPCN[UKU PQVCVKQP HQT UWO
6CTUMK OQFGNU
EQORTGJGPUKQP
6CTUMK #NHTGF
5GG WPFGT
OQFGNU
UKORNG RTGFKECVKXG
5GG
WPFGT RTGFKECVKXG
6GPPCPV 0GKN KZ
UKORNG
5GG
\KI\CI VJGQT[
VJKTFQTFGT NQIKE
XU
EQORNGZ
VJKTFQTFGT 2GCPQ CTKVJOGVKE
5KORUQP 5VGRJGP 5MQNGO CTKVJOGVKE
5GG
5GG
2GCPQ
CTKVJOGVKE VJKTFQTFGT
RTKOKVKXG TGEWTUKXG
VJQWIJV
CTKVJOGVKE 24#
VTCPUHKPKVG ECTFKPCNU CPF QTFKPCNU
5MQNGO RCTCFQZ P
UOCNN CPF XGT[ UOCNN 5QNQXC[ 4QDGTV
VTCPUKVKXG ENQUWTG
UQTVCN RTGFKECVG
VTCPUKVKXG UGV
URNKVVKPI
VTCPUKVKXKV[ CPF VTCPUKVKXG ENQUWTG
UVCPFCTF OQFGNU
RTQQH
Σ Σ KP CTKVJOGVKE Σ Σ
KP UGV VJGQT[
XU
GZVGPFGF
5JQGPHKGNF ,QUGRJ P P
Σ
OQFGN VJGQT[
5\OKGNGY6CTUMK VJGQTGO FQWDN[ GZVGPFGF
P JKUVQT[ QH P
5GG
5\OKGNGY 9CPFC
5JQGPHKGNFOU VJGQTGO
5KIOC HQTOWNCU
UGOCPVKE
XU
5GG
VTKEJQVQO[
OQFGNU
5VGKPGT /CTM P
VTWVJXCNWGU P
UVTWEVWTCNKUO
V[RGU VJGQT[ QH UKORNG
UWDQTFKPCVKQP CZKQOU QH RTKPEKRNG QH
XU
XU
TCOKHKGF
UGV VJGQT[
UWEEGUUQT FGHKPKVKQP QH PQVCVKQP CPF VJGQTGO UWK IGPGTKU
XKGY QH PWODGTU
HQT
WNVTCRTGFKECVKXG WPKSWG GZKUVGPEG
5GG WPFGT
WPKXGTUCN EQPEGRV 7 UGV 8
UWKVCDKNKV[ UVTQPI YGCM
WPKXGTUCN SWCPVKHKGT
UWO
WPUCVWTCVGF
5GG
CFFKVKQP
UWOOCDKNKV[
XU
CFFKVKXKV[
SWCPVKHKGTU
5GG
∀ 5GG WPFGT SWCPVKHKGT
UCVWTCVGF
XU
7TSWJCTV #NCUFCKT P
WPUCVWTCVGF
8
5GG
.CY 8 WPKXGTUCN UGV
XCNKFKV[
XU
FGOQPUVTCDKNKV[
XCTKCDNGDKPFKPI VGTOHQTOKPI QRGTCVQTU EQPEGRV n UGV ] _ PWODGT ^^ ^^ 8QP 0GWOCPPOU CZKQO XQP 0GWOCPP $GTPC[U )ÑFGN UGV VJGQT[ 0$) XQP 0GWOCPP ,QJP FGHKPKVKQP QH PWODGT JKU CZKQOCVK\CVKQP QH UGV VJGQT[
5GG CNUQ
XQP 0GWOCPP
$GTPC[U )ÑFGN UGV VJGQT[
9CPI *CQ P P 9GJOGKGT -CK KZ P P 9GKPGT-WTCVQYUMK FGHKPKVKQP
5GG
QTFGTGF RCKT
YGNNQTFGT 9JKVGJGCF # 0 9KNMKG #NGZ 9TKIJV %TKURKP KZ
5GG
UGRCTCVKQP CZKQO
VJGQT[
5GG CNUQ
\GTQ FGHKPKVKQP QH \GTQRNCEG U[ODQNU \KI\CI VJGQT[
UGRCTCVKQP
