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IVB?J:JOBYLBIH< Lbihij_^_ey_lkydZdh[eZklvagZqbfhklbijhihabpbhgZevghcnmgdpbbl_dZdkh\h dmighklv Zj]mf_glh\ ^ey dhlhjuo mdZaZggZy nmgdpby bf__l agZq_gby
GZijbf_j, ImZgdZj_. Kf.: Revue de Métaphysique et de Morale (May, 1906).
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Kf Principles of Mathematics, § 48.
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qbkehlhq_d1< o).φo ogZau\Z_lkyfgbfhci_j_f_gghc dh]^Z φoml\_j`^Z_lkylZf]^_ og_mlhqgzgogZau\Z_lky^_ckl\bl_evghci_j_f_gghc. DZdZy-lh ij_^bdZlb\gZy nmgdpby hl Zj]mf_glZ dZdh]h-lh lbiZ ih h[klhyl_evkl \Zf hgZ [m^_l ij_^klZ\e_gZ dZd φ!o, φ!α beb φ!R Ij_^bdZlb\gZy nmgdpby hl o – wlh nmgdpby qvb agZq_gby y\eyxlky ijhihabpbyfb hlghkysbfbky d lbim ke_^mxs_fm aZ lbihf o _keb o y\ey_lkybg^b\b^hfbeb ijhihabpb_c beb aZ lbihf agZq_gbc o _keb o y\ey_lkynmgdpb_cHgZfh`_l[ulvhibkZgZdZdnmgdpby\dhlhjhc\k_fgbfu_i_j_ f_ggu__keblZdh\u__klvhlghkylkydh^ghfmlbimkobebdf_gvr_fmlbimI_j_f_g gZyhlghkblkydf_gvr_fmq_folbim_kebhgZfh`_lagZqbfh\klj_qZlvkydZdZj]mf_gl \kZfhf obebdZdZj]mf_gl\Zj]mf_gl_kZfh]hobl^ Ml\_j`^_gb_ l_ml\_j`^_gb_qlhg_dhlhjZyijhihabpbyy\ey_lkybklbgghcbeb qlh dZdh_-lh agZq_gb_ g_dhlhjhc ijhihabpbhgZevghc nmgdpbb y\ey_lky bklbgguf Ml \_j`^_gb_ lj_[m_lky ^ey lh]h qlh[u hlebqblv ^_ckl\bl_evgh ml\_j`^Z_fmx ijhihab pbxhlijhihabpbbijhklhjZkkfZljb\Z_fhcbebhlijhihabpbbgZdhlhjmxkkueZxlky dZdgZmkeh\b_g_dhlhjhc^jm]hcijhihabpbbGZml\_j`^_gb_[m^_lmdZau\ZlvagZdµL¶ ij_^ihkeZgguc lhfm qlh ml\_j`^Z_lky k ^hklZlhqguf dhebq_kl\hf lhq_d qlh[u aZ dexqblvlhqlhml\_j`^Z_lky\kdh[db2. I_j_^ l_f dZd i_j_clb d bkoh^guf ijhihabpbyf gZf gm`gu g_dhlhju_ hij_^_e_ gby<ke_^mxsbohij_^_e_gbyolZd`_dZdb\bkoh^guoijhihabpbyo[md\up, q, rbk ihevamxlky^eyh[hagZq_gbyijhihabpbc p ⊃ q . = . ∼p ∨ q Df. Wlhhij_^_e_gb_mklZgZ\eb\Z_lqlhµp ⊃ q¶ dhlhjh_ijhqblu\Z_lkydZdµj\e_qzlq’) ^he`ghhagZqZlvµj –eh`ghbebq –bklbggh¶Yg_gZf_j_\Zxkvml\_j`^Zlvqlhµ\e_ qzl¶g_fh`_lbf_lv^jm]h]hkfukeZghml\_j`^ZxlhevdhlhqlhwlhlkfukegZb[he__ ih^oh^bl ^ey lh]h qlh[u aZ^Zlv µ\e_qzl¶ \ kbf\hebq_kdhc eh]bd_ < hij_^_e_gbb agZd jZ\_gkl\Z b [md\u µDf¶ ^he`gu jZkkfZljb\Zlvky dZd h^bg kbf\he kh\f_klgh hagZqZy µagZqblihhij_^_e_gbx¶AgZdjZ\_gkl\Z[_a[md\µDf¶bf__lbghckfukedhlhjuc\kdh j_[m^_ljZkkfhlj_g p . q . = . ∼(∼p ∨ ∼q)
Df.
Wlhhij_^_ey_leh]bq_kdh_ijhba\_^_gb_^\moijhihabpbcjbql_µjbqh[Zy\ey xlky bklbggufb¶ Ijb\_^zggh_ hij_^_e_gb_ mklZgZ\eb\Z_l qlh wlh ^he`gh hagZqZlv µEh`ghqlhj –eh`ghbebq –eh`gh¶A^_kvhij_^_e_gb_kgh\Zg_^Zzl_^bgkl\_ggh]h kfukeZdhlhjucfh`_l [ulvijb^Zgµjbqh[Zy\eyxlkybklbggufb¶ gh aZ^Zzl agZq_ gb_dhlhjh_gZb[he__ih^oh^bl^eygZr_cp_eb p ≡ q . = . p ⊃ q . q ⊃ p Df.
Ijbbkihevah\Zgbblhq_dfuke_^m_fI_ZghWlh bkihevah\Zgb_ iheghklvx h[tykg_gh f-jhf MZclo_ ^hf; kf.: ‘On Cardinal Numbers’, American Journal of Mathematics, Vol, XXIV, b ‘On Mathematical Concepts of Material World’, Phil. Trans. A., Vol. CCV, P. 472. 2 WlbfagZdhfdZdb\\_^_gb_fb^_bdhlhjmxhg\ujZ`Z_lfuh[yaZguNj_]_Kf_]hBegriffsschrift (Halle, 1879), C>Jmkkdbci_j_\h^kf Bkqbke_gb_ihgylbcNj_]_=Eh]bdZbeh]bq_kdZyk_fZglb dZ–F :ki_dlIj_kk@bGrundgesetze der Arithmetik (Jena, 1983), Vol. I, C. 9. 1
Lh _klv µp ≡ q¶ dhlhjh_ qblZ_lky dZd µj wd\b\Ze_glgh q¶ hagZqZ_l µj \e_qzl q b q \e_qzlj¶ hldm^Zdhg_qghke_^m_lqlhjbqy\eyxlkyh[Zbklbggufbbebh[Zeh`gufb (∃ ∃o) . φo . = . ∼{(x) . ∼φx}
Df.
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Df.
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Df,
bl^^eyex[h]hqbkeZi_j_f_gguo Lj_[mxlky ke_^mxsb_ bkoh^gu_ ijhihabpbb \ b p, q, r h[hagZqZxl ijhihabpbb (1) Ijhihabpby\u\_^_ggZybabklbgghcihkuedby\ey_lkybklbgghc (2) L: p ∨ p . ⊃ . p. (3) L: q . ⊃ . p ∨ q. (4) L: p ∨ q . ⊃ . q ∨ p. (5) L: p ∨ (q ∨ r) . ⊃ . q ∨ (p ∨ r). (6) L: . q ⊃ r . ⊃ : p ∨ q . ⊃ . p ∨ r. (7) L: (x) . φx . ⊃ . φy; l_µ_keb\k_agZq_gbyφ ∧o y\eyxlkybklbggufblhφmy\ey_lkybklbgguf]^_φm_klvdZ dh_-lhagZq_gb_¶1. ?keb φm – bklbggh ]^_ φm _klv dZdh_-lh agZq_gb_ φ ∧o lh o).φo – bklbggh Wlh]h g_evay\ujZablv\gZrbokbf\heZo b[h_kebfuaZibku\Z_fµφm . ⊃ . (o)φo¶wlhhagZqZ_l ‘φm\e_qzlqlh\k_agZq_gby φ ∧o y\eyxlkybklbggufb]^_mfh`_lijbgbfZlvex[h_agZ q_gb_ih^oh^ys_]hlbiZ¶qlh\h[s_fg_bf__lf_klZLhqlhfugZf_j_\Z_fkyml\_j `^ZlvaZdexqZ_lky\ke_^mxs_f µ?kebijbex[hf\u[jZgghfm φm –bklbgghlh o).φo – bklbggh¶lh]^ZdZdlhqlh\ujZ`_ghihkj_^kl\hfµφy . ⊃ . (x) . φx¶_klvµIjbex[hf\u
1
M^h[ghbkihevah\ZlvaZibkvφoqlh[uh[hagZqblvkZfmnmgdpbx\ijhlb\hiheh`ghklvlhfmbebbgh fmagZq_gbxwlhcnmgdpbb
[jZgghfm_kebφm –bklbgghlh o).φo –bklbggh¶qlhy\ey_lkykh\_jr_gghbguf\ukdZ au\Zgb_fdhlhjh_\h[s_fkemqZ_eh`gh (9) L: (o) . φo . ⊃ . φZ]^_Z_klvdZdZy-lhhij_^_ezggZydhgklZglZ WlhijbgpbigZkZfhf^_e_ij_^klZ\ey_lkh[hcfgh]hjZaebqguoijbgpbih\Zbf_g gh klhevdh kdhevdh kms_kl\m_l \hafh`guo agZq_gbc Z L_ hg mklZgZ\eb\Z_l gZijb f_j qlh lh qlh bf__l kbem ^ey \k_o bg^b\b^h\ bf__l kbem ^ey KhdjZlZ Z lZd`_ hgh bf__l kbem ^ey IeZlhgZ b l^ Wlhl ijbgpbi khklhbl \ lhf qlhh[s__ijZ\beh fh`gh ijbf_gblvdqZklghfmkemqZx ghqlh[uaZ^Zlv_]hh[eZklvg_h[oh^bfhmihfygmlvhl ^_evgu_ijbf_juihkdhevdm\ijhlb\ghfkemqZ_gZfgm`_gijbgpbidhlhjuckZfaZ\_ jylgZk\h[s_fijZ\be_qlhh[sb_ijZ\beZdhlhju_fh]mlijbf_g_gudqZklghfmkem qZx fh]ml [ulv ijbf_g_gu d hl^_evghfm kemqZx kdZ`_f d KhdjZlm LZdbf h[jZahf wlhlijbgpbihlebqZ_lkyhl ^Zggucijbgpbi\ukdZau\Z_lkyhKhdjZl_IeZlhg_beb dZdhc-lh^jm]hcdhgklZgl_lh]^ZdZd \ukdZau\Z_lkyhi_j_f_gghc MdZaZgguc ijbgpbi gbdh]^Z g_ bkihevam_lky \ kbf\hebq_kdhc eh]bd_ beb \ qbklhc fZl_fZlbd_ihkdhevdm\k_gZrbijhihabpbby\eyxlkyh[sbfbB^Z`_lh]^Zdh]^Z dZd \µh^bg_klvqbkeh¶ fuih\b^bfhklbbf__fkljh]hqZklguckemqZcijb[ebadhfjZk kfhlj_gbbhgg_hdZau\Z_lkylZdh\ufNZdlbq_kdbijbf_g_gb_wlh]hijbgpbiZy\ey_l ky hlebqbl_evguf ijbagZdhf ijbdeZ^ghc fZl_fZlbdb KlZeh [ulv kljh]h ]h\hjy fu ^he`gubkdexqblv_]hbagZr_]hkibkdZ (10) L: . (o) . j ∨ φo . ⊃ : j . ∨ . (o) . φo; l_µ_keb³jbebφo” –\k_]^Zbklbgghlhbebj –bklbgghbebφo –\k_]^Zbklbggh¶ Dh]^Zf(φx) –bklbgghijbex[hf\hafh`ghfZj]mf_gl_obF(φy) –bklbgghijb ex[hf\hafh`ghfZj]mf_gl_mlh]^Z^f(φx) . F(φx `y\ey_lkybklbggufijbex[hf\ha fh`ghfZj]mf_gl_o. Wlh–ZdkbhfZµg_hij_^_ezgghklbi_j_f_gguo¶HgZgm`gZdh]^ZhdZ`^hcba^\mo hl^_evguoijhihabpbhgZevguonmgdpbcba\_klghqlhhgb\k_]^Zy\eyxlkybklbggufb b fu ohlbf \u\_klb qlh bo eh]bq_kdh_ ijhba\_^_gb_ \k_]^Z y\ey_lky bklbgguf Wlhl \u\h^ hijZ\^Zg lhevdh lh]^Z dh]^Z ^\_ nmgdpbb ijbgbfZxl Zj]mf_glu h^gh]h b lh]h `_lbiZb[h\ijhlb\ghfkemqZ_boeh]bq_kdh_ijhba\_^_gb_[_kkfuke_ggh ?keb φo.φo⊃ψo –bklbggh^eyex[h]h\hafh`gh]holh ψo –bklbggh^eyex[h]h \hafh`gh]ho. WlZ ZdkbhfZ lj_[m_lky ^ey lh]h qlh[u aZ\_jblv gZk \ lhf qlh h[eZklv agZqbfhklb ψo\ij_^iheZ]Z_fhfkemqZ_kh\iZ^Z_lkh[eZklvxagZqbfhklbφo.φo⊃ψo.⊃.ψo nZdlbq_ kdbh[_h[eZklbkh\iZ^Zxlkh[eZklvxagZqbfhklbφo<ij_^iheZ]Z_fhfkemqZ_fuagZ _f qlh ψo – bklbggh \_a^_ ]^_ b φo.φo⊃ψo b φo.φo⊃ψo.⊃.ψo y\eyxlky agZqbfufb gh [_aZdkbhfufug_agZ_fqlh ψo –bklbggh\_a^_]^_ ψoy\ey_lkyagZqbfufKe_^h\Z l_evghwlZZdkbhfZgZfg_h[oh^bfZ :dkbhfu b lj_[mxlkygZijbf_jijb^hdZaZl_evkl\_ (o) . φo : (o) . φo ⊃ ψo : ⊃ . (o) . ψo. Ih b L: . (o) . φo : (o) . φo ⊃ ψo : ⊃ : φm . φm ⊃ ψm, hlkx^Zih L: . (o) . φo : (o) . φo ⊃ ψo : ⊃ : ψm,
hlkx^Zj_amevlZl\ul_dZ_lih b (13) L: . (∃ ∃f) : . (x) : φx . ≡ . f!x. Wlh – ZdkbhfZ k\h^bfhklb HgZ mklZgZ\eb\Z_l qlh _keb aZ^Zlv dZdmx-lh nmgdpbx ∧ φ o lhkms_kl\m_llZdZyij_^bdZlb\gZynmgdpbyf! ∧o qlhf!x\k_]^Zwd\b\Ze_glgZ φoAZ ∃f ¶ihhij_^_e_gbx_klvhljbpZgb_ f_lbfqlhihkdhevdmijhihabpbygZqbgZxsZykykµ ∃ ijhihabpbbgZqbgZxs_ckykµ f ¶ijb\_^zggZyZdkbhfZ\dexqZ_l\hafh`ghklvjZkkfhl j_gby µ\k_o ij_^bdZlb\guo nmgdpbc hl o¶ ?keb φo _klv dZdZy-lh nmgdpby hl o fu g_ ∃φ ¶ihkdhevdmfug_fh`_f fh`_f\ukdZaZlvijhihabpbxgZqbgZxsmxkykµ φ ¶bebµ ∃ jZkkfZljb\Zlv µ\k_ nmgdpbb¶ gh lhevdh µdZdmx-lh nmgdpbx¶ beb µ\k_ ij_^bdZlb\gu_ nmgdpbb¶ (14) L: . (∃ ∃f) : . (x, y) : φ(x, y) . ≡ . f!(x, y). Wlh–ZdkbhfZk\h^bfhklb^ey^\mof_klghcnmgdpbb < ijb\_^zgguo \ur_ ijhihabpbyo gZrb o b m fh]ml hlghkblvky d ex[hfm lbim ?^bgkl\_ggh_ ]^_ mf_klgZ l_hjby lbih\ khklhbl \ lhf qlh ebrv iha\hey_l gZf hlh`^_kl\blv ^_ckl\bl_evgu_ i_j_f_ggu_ \klj_qZxsb_ky \ jZaebqguo kh^_j`Zgbyo dh]^Z^_fhgkljbjm_lkyqlhhgbhlghkylkydh^ghfmblhfm`_lbimihkdhevdm\h[hbo kemqZyo\oh^yldZdZj]mf_gluh^ghcblhc`_nmgpbbbqlh\ b mbZkhhl\_lkl \_ggh^he`guhlghkblkydlbimih^oh^ys_fm^eyZj]mf_glh\ φ ∧z Ihwlhfm_kebij_^ iheh`blvgZijbf_jqlhmgZk_klvijhihabpbynhjfu φ).f!(φ! ∧z , x y\eyxsZyky\lhjh ihjy^dh\hcnmgdpb_chlolhih L: (φ) . f!(φ! ∧z , x) . ⊃ . f!(ψ! ∧z , x), ]^_ ψ! ∧z _klvdZdZy-lhnmgdpbyi_j\h]hihjy^dZGh φ) . f!(φ! ∧z , x g_evayjZkkfZljb\Zlv lZddZd_keb[uhgZ[ueZi_j\hihjy^dh\hcnmgdpb_chlob[jZlvwlmnmgdpbxdZd\ha fh`gh_agZq_gb_ ψ! ∧z \mdZaZgghf\ur_\ujZ`_gbbIh^h[gh_kf_r_gb_lbih\ijb\h ^bldiZjZ^hdkme`_pZ. Kgh\Z jZkkfhljbf deZkku dhlhju_ g_ y\eyxlky qe_gZfb kZfbo k_[y Ykgh qlh ih kdhevdmfuhlh`^_kl\ey_fdeZkkuknmgdpbyfb1gbh[h^ghfdeZkk_g_evayagZqbfh]h \hjblvqlhhgy\ey_lkybebg_y\ey_lkyqe_ghfkZfh]hk_[y b[hqe_gudeZkkZy\eyxlky Zj]mf_glZfb nmgdpbb Z Zj]mf_glu nmgdpbb \k_]^Z hlghkylky d lbim [he__ gbadhfm q_fnmgdpbyB_kebfukijhkbf µDZdh[klhbl^_ehkdeZkkhf\k_odeZkkh\"Hgqlh`_ g_ y\ey_lky deZkkhf b ihwlhfm qe_ghf kZfh]h k_[y"¶ hl\_l ^\hckl\_g_g
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VIIWE?F?GL:JG:YL?HJBYDE:KKH<BHLGHR?GBC Ijhihabpbb\dhlhju_\oh^blnmgdpby φfh]mlihk\h_fmbklbgghklghfmagZq_gbx aZ\bk_lvhlhkh[hcnmgdpbbφbeb`_hgbfh]mlaZ\bk_lvhlh[tzfZφl_hlZj]mf_glh\ dhlhju_\uihegyxl φNmgdpbbihke_^g_]hkhjlZfu[m^_fgZau\Zlvwdkl_gkbhgZevgu fbLZdgZijbf_jµY\_jxqlh\k_ex^bkf_jlgu¶g_fh`_l[ulvwd\b\Ze_glghµY\_ jxqlh\k_[_kizju_^\mgh]b_kf_jlgu¶^Z`__kebex^bihh[tzfmkh\iZ^Zxlk^\mgh ]bfb [_kizjufb b[h y fh]m b g_ agZlv qlh ih h[tzfm hgb h^bgZdh\u Gh µ
Df.
Nmgdpbyf{ ∧z (ψz `nZdlbq_kdb_klvnmgdpbyhl ψ ∧z ohlyhgZbg_kh\iZ^Z_lknmgdpb_c f(ψ! ∧z ij_^iheZ]Zy qlh wlZ ihke_^gyy y\ey_lky agZqbfhc Gh ljZdlh\Zlv lZd f{ ∧z (ψz)} l_ogbq_kdb m^h[gh ohly hgZb kh^_j`bl Zj]mf_gl ∧z (ψz dhlhjuc fugZau\Z_fµdeZkk hij_^_ey_fucihkj_^kl\hfψ¶Fubf__f L: . φx . ≡x . ψx : ⊃ : f{ ∧z (φz)} . ≡ . f{ ∧z (ψz)},
ke_^h\Zl_evgh ijbf_gyy hij_^_e_gb_ lh`^_kl\Z d nbdlb\guf h[t_dlZf ∧z (φz b ∧z (ψz), ^Zggh_\ur_fugZoh^bfqlh L: . φx . ≡x . ψx : ⊃ . ∧z (φz) = ∧z (ψz).
Wlhml\_j`^_gb_ZlZd`__]hdhg\_jkby qlhlZd`_fh`gh^hdZaZlv mdZau\Z_lhleb qbl_evgh_ k\hckl\h deZkkh\ Ke_^h\Zl_evgh fu \iheg_ fh`_f ljZdlh\Zlv ∧z (φz dZd deZkkhij_^_ey_fucihkj_^kl\hfφL_f`_kZfufkihkh[hffumklZgZ\eb\Z_f f{ ∧x ∧y ψ(x, y)} . = : (∃ ∃φ) : φ!(x, y) . ≡x, y . ψ(x, y) : f{φ!( ∧x , ∧y )}
Df.
A^_kvg_h[oh^bfhg_kdhevdhkeh\hlghkbl_evghjZaebqbyf_`^m φ!( ∧x , ∧y b φ!( ∧y , ∧x Fu [m^_f ijbgbfZlv ke_^mxs__ kh]eZr_gb_ Dh]^Z nmgdpby \ ijhlb\hiheh`ghklv k\hbf agZq_gbyf ij_^klZ\e_gZ \ nhjf_ \dexqZxs_c ∧x b ∧y beb dZdb_-lh ^jm]b_ ^\_ [md\u ZenZ\blZ agZq_gb_wlhcnmgdpbb^eyZj]mf_glh\Zbb^he`ghh[gZjm`b\Zlvkyih^klZ gh\dhc Z \f_klh ∧x b b \f_klh ∧y l_ Zj]mf_gl mihfbgZxsbcky i_j\uf ^he`_g ih^ klZ\eylvky \f_klh [md\u dhlhjZy \klj_qZ_lky \ ZenZ\bl_ jZgvr_ Z Zj]mf_gl mihfb
gZxsbcky \lhjuf – \f_klh [md\u dhlhjZy \klj_qZ_lky iha^g__ B wlh \iheg_ m^h\e_ l\hjbl_evghijh\h^bljZaebqb_f_`^mφ!( ∧x , ∧y bφ!( ∧y , ∧x gZijbf_j AgZq_gb_φ!( ∧x , AgZq_gb_φ!( ∧x , AgZq_gb_φ!( ∧y , AgZq_gb_φ!( ∧y , FumklZgZ\eb\Z_f ke_^h\Zl_evgh
∧ y ^eyZj]mf_glh\Zbb_klvφ!(Z, ∧ y ^eyZj]mf_glh\bbZ_klvφ!(b, ∧ x ^eyZj]mf_glh\Zbb_klvφ!(b, ∧ x ^eyZj]mf_glh\bba_klvφ!(a,
o∈φ! ∧z . = . φ!o
b). Z). a). b).
Df.,
L: . x∈ ∧z (ψz) . = : (∃ ∃φ) : φ!y . ≡y . ψy : φ!x.
Dlhfm`_ihZdkbhf_k\h^bfhklbfubf__f
(∃ ∃φ) : φ!y . ≡y . ψy,
ke_^h\Zl_evgh
L: x∈ ∧z (ψz) . ≡ . ψx.
Wlh bf__l kbem ijb ex[hf o Ij_^iheh`bf l_i_jv qlh fu ohlbf jZkkfhlj_lv
∧ ∧ ∧ z (ψz)∈ φ f{ z (φ!z `Kh]eZkghbaeh`_gghfm\ur_fubf__f
hlkx^Z
L: . ∧z (ψz)∈ φ∧ f{ ∧z (φ!z)} . ≡ . f{ ∧z (ψz)} : ≡ : (∃ ∃φ) : φ!y . ≡y . ψy : f(φ!z), L: . ∧z (ψz) = ∧z (χz) . ⊃ : ∧z (ψz)∈x . ≡χ . ∧z (χz)∈x,
]^_oaZibku\Z_lky\f_klhex[h]h\ujZ`_gbynhjfu φ∧ f{ ∧z (φ!z)}. FumklZgZ\eb\Z_f Df. ∃φ} . α = ∧z (φ!z)} cls = α∧ {(∃
A^_kv cls h[eZ^Z_l agZq_gb_f dhlhjh_aZ\bkblhl lbiZ fgbfhc i_j_f_gghc φ Ke_^h\Z l_evgh ijhihabpby µcls ∈ cls¶ gZijbf_j y\eyxsZyky ke_^kl\b_f ijb\_^zggh]h \ur_ hij_^_e_gbylj_[m_lqlhµcls’^he`ghh[eZ^ZlvjZaebqgufagZq_gb_f\^\mof_klZo]^_ hgh \klj_qZ_lky Kbf\he µcls’ fh`_l bkihevah\Zlvky lhevdh lZf ]^_ g_h[oh^bfh agZlv lbi hgh[eZ^Z_lg_hij_^_ezgghklvxdhlhjZyijbkihkZ[eb\Z_lkydh[klhyl_evkl\Zf?k eb fu \\h^bf dZd g_hij_^_ey_fmx nmgdpbx µIndiv!x’ hagZqZxsmx µx – bg^b\b^¶ fu fh`_fmklZgh\blv Kl =
∧ ∃φ} α {(∃
. α = ∧z (φ!z . Indiv!z)}
Df.
Lh]^ZKl –wlhhij_^_ezgguckbf\hehagZqZxsbcµdeZkkbg^b\b^h\¶ Fu[m^_fbkihevah\Zlvkljhqgu_[md\u]j_q_kdh]hZenZ\blZ bgu_q_f ∈, φ, ψ, χ, θ qlh[uij_^klZ\eylvdeZkkuex[h]hlbiZl_h[hagZqZlvkbf\heunhjfu ∧z (φ!z beb ∧ z (φz). Kwlh]himgdlZl_hjbydeZkkh\\hfgh]hfjZa\b\Z_lkydZd\kbkl_f_I_Zgh ∧z (φz aZ f_gy_lz%(φz LZd`_ymklZgZ\eb\Zx
α ⊂ β . = : x∈α . ⊃ . x∈β Df., Df., ∃!α . = . (∃ ∃x) . x∈α ∧ V = x (x = x)Df., Λ = ∧x {∼(x = x)} Df., ∃, Λ, VdZdbkbf\heuclsb ∈g_hij_ ]^_ΛdZdbmI_Zgh_klvgmev-deZkkKbf\heu∃ ^_e_gubijbh[j_lZxlhij_^_ezggh_agZq_gb_dh]^ZjZkkfZljb\Z_fuclbimdZaZgbguf kihkh[hf Hlghr_gbyfuljZdlm_flhqghlZdbf`_kihkh[hfmklZgZ\eb\Zy a{φ!( ∧x , ∧y )}b . = . φ!(a, b)
Df.
ihjy^hd ij_^hij_^_ezg ZenZ\blguf ihjy^dhf o b m b lbih]jZnkdbf ihjy^dhf Z b b); hlkx^Z L: . a{ ∧x ∧y ψ(x, y)}b . ≡ : (∃ ∃φ) : ψ(x, y) . ≡x, y . φ!(x, y) : φ!(a, b), hldm^ZihZdkbhf_k\h^bfhklb L: a{ ∧x ∧y ψ(x, y)}b . ≡ . ψ(a, b). Bkihevamyijhibkgu_[md\ueZlbgkdh]hZenZ\blZ\dZq_kl\_khdjZs_gby^eylZdbokbf \heh\dZd ∧x ∧y ψ(x, y fugZoh^bfqlh ]^_ FumklZgZ\eb\Z_f
L: . R = S . ≡ : xRy . ≡x, y . xSy, R = S . = : f!R . ⊃f . f!S Rel =
∧ R {∃ ∃φ)
.R=
∧ ∧ x y φ!(x,
Df. y)} Df.
b gZoh^bf qlh \kz qlh ^hdZau\Z_lky ^ey deZkkh\ bf__l k\hc ZgZeh] ^ey ^\mof_klguo hlghr_gbcKe_^myI_ZghfumklZgZ\eb\Z_f
α∩β = ∧x (x∈α . x∈β)
Df.,
hij_^_eyyijhba\_^_gb_bebh[smxqZklv^\modeZkkh\
α∪β = ∧x (x∈α . ∨ . x∈β) Df., hij_^_eyykmffm^\modeZkkh\ b – α = ∧x {∼(x∈α)} Df., hij_^_eyyhljbpZgb_deZkkZKoh^gufh[jZahf^eyhlghr_gbcfumklZgZ\eb\Z_f
•
R ∩S = •
R∪S =
∧ ∧ x y (xRy
∧ ∧ x y (xRy
. xSy)
Df.,
. ∨ . xSy)
• ∧ − R = x ∧y {∼(xRy)}
Df., Df.
VIII. >?KDJBILB
pbclZdbodZdR ∩ S[uebijhihabpbhgZevgufbGhh[uqgu_nmgdpbbfZl_fZlbdblZ db_dZdo2, sin x, log xg_y\eyxlkyijhihabpbhgZevgufbNmgdpbbwlh]h\b^Z\k_]^Zha gZqZxlµwe_f_glbf_xsbclZdh_-lhblZdh_-lhhlghr_gb_do¶Ihwlhcijbqbg_hgbfh ]ml [ulv gZa\Zgu ^_kdjbilb\gufb [descriptive@ nmgdpbyfb ihkdhevdm hgb hibku\Zxl [describe@ hij_^_ezgguc we_f_gl q_j_a _]h hlghr_gb_ d bo Zj]mf_glZf LZd µsin π/2’ hibku\Z_lqbkeh h^gZdhijhihabpbb\dhlhjuo\klj_qZ_lky πg_hklZgmlkyl_fb`_ kZfufb_keb[u\gbo[uehih^klZ\e_ghWlhgZijbf_jh[gZjm`b\Z_lkybaijhihab pbbµsin π ¶dhlhjZykh^_j`blagZqbfmxbgnhjfZpbxlh]^ZdZdµ ¶–ljb\bZevgh >_kdjbilb\gu_ nmgdpbb bf_xl agZq_gb_ g_ kZfb ih k_[_ gh lhevdh dZd dhgklblm_glu ijhihabpbc bwlh\hh[s_ijbf_gy_lkydnjZaZfnhjfuµwe_f_glbf_xsbclZdh_-lhb lZdh_-lhk\hckl\h¶Ke_^h\Zl_evghbf_y^_ehklZdbfbnjZaZfbfu^he`guhij_^_eylv dZdmx-lhijhihabpbx\dhlhjmxhgb\oh^ylZg_njZamkZfmihk_[_1LZdbfh[jZahf fu ijboh^bf d ke_^mxs_fm hij_^_e_gbx \ dhlhjhf µ x)(φx)’ ^he`gh qblZlvky dZd ‘^Zgguc [the@we_f_glxdhlhjuc\uihegy_lφo’. •
∃b) : φx . =x . x=b : ψb ψ{( x)(φx)} . = : (∃
Df.
Wlh hij_^_e_gb_ mklZgZ\eb\Z_l qlh µwe_f_gl dhlhjuc \uihegy_l φ \uihegy_l ψ’ ^he`gh hagZqZlv µKms_kl\m_l l_jfbg b lZdhc qlh φo – bklbggh lh]^Z b lhevdh lh]^Z dh]^Zo_klvbb ψb –bklbggh¶LZdbfh[jZahf\k_ijhihabpbbh[µ^ZgghflZdhf-lhb lZdhf-lh¶[m^mleh`gufb_keblZdh]h-lhblZdh]h-lhg_kms_kl\m_lbebbokms_kl\m_l g_kdhevdh H[s__hij_^_e_gb_^_kdjbilb\ghcnmgdpbby\ey_lkyke_^mxsbf R‘y = (
x)(xRy) Df.;
l_µR‘y¶^he`ghhagZqZlvµwe_f_gldhlhjucbf__lhlghr_gb_Rdm¶?keb`_kms_kl\m _lg_kdhevdhbebg_kms_kl\m_lgbh^gh]hwe_f_glZbf_xs_]hhlghr_gb_Rdmlh\k_ ijhihabpbbhR‘y[m^mleh`gufbFumklZgZ\eb\Z_f E!(
x)(φx) . = : (∃ ∃b) : φx . ≡x . x=b
Df.
A^_kvµE!( x)(φx ¶fh`_lijhqblu\ZlvkyµKms_kl\m_llZdhcwe_f_gldZdodhlhjuc\u ihegy_lφo¶bebµlhlodhlhjuc\uihegy_lφokms_kl\m_l¶Fubf__f L: . E!R‘y . ≡ : (∃ ∃b) : xRy . ≡x . x=b. 1
Kfmihfygmlmx\ur_klZlvxµOn Denoting¶]^_ijbqbguwlh]hij_^klZ\e_gu[he__ijhkljZggh
DZ\uqdZ\R‘yfh`_lijhqblu\ZlvkyLZd_kebR –hlghr_gb_hlpZdkugmlhµR‘y¶_klv µhl_pm¶?kebR –hlghr_gb_kugZdhlpm\k_ijhihabpbbhR‘y[m^mleh`gufb_kebm g_bf__lgbh^gh]hbeb[hevr_q_fh^gh]hkugZ BakdZaZggh]h\ur_h[gZjm`b\Z_lkyqlh^_kdjbilb\gu_nmgdpbbihemqZxlkybahl ghr_gbc Hij_^_ey_fu_ l_i_jv hlghr_gby ]eZ\guf h[jZahf \Z`gu ^ey jZkkfhlj_gby ^_kdjbilb\guonmgdpbcdhlhjufhgb^ZxlgZqZeh ∧
∧
Cnv = Q P {xQy . ≡x, y . yPx}
Df.
A^_kvCnv_klvkhdjZs_gb_^eyµdhg\_jkby¶Wlhhij_^_ey_lhlghr_gb_g_dh]hhlghr_ gby d k\h_c dhg\_jkbb gZijbf_jhlghr_gb_ hlghr_gby[hevr_ dhlghr_gbx f_gvr_, hlghr_gby hlph\kl\Z d hlghr_gbx kugh\kl\Z hlghr_gb_ ij_^r_kl\_ggbdZ d hlghr_ gbxgZke_^gbdZbl^Fubf__f L. Cnv‘P = ( Q){xQy . ≡x, y . yPx}.
>eykhdjZs_gbyaZibkbqlhqZklh[he__m^h[ghfumklZgZ\eb\Z_f ∪
P = Cnv‘P Df.
GZflj_[m_lky_szh^gZaZibkv^eydeZkkZl_jfbgh\bf_xsbohlghr_gb_RdmKwlhc p_evxfumklZgZ\eb\Z_f → ∧ R = α ∧y {α = ∧x (xRy)}
hlkx^Z
Df.,
L. R ‘y = ∧x (xRy). →
Koh^gufh[jZahffumklZgZ\eb\Z_f
← ∧ R = β ∧x {β = ∧y (xRy)}
hlkx^Z
Df.,
L. R ‘x = ∧y (xRy). ←
>Ze__gZflj_[m_lkyh[eZklvR l_deZkkwe_f_glh\bf_xsbohlghr_gb_Rdq_fmeb[h dhg\_jkgZyh[eZklvR l_deZkkwe_f_glh\ddhlhjufqlh-eb[hbf__lhlghr_gb_ R bihe_ Rij_^klZ\eyxs__kh[hckmffmh[eZklbRbdhg\_jkghch[eZklbRKwlhcp_ evx fu hij_^_ey_f hlghr_gby h[eZklb dhg\_jkghc h[eZklb b ihey d R Hij_^_e_gby lZdh\u ∧
∧
∃y) . xRy)} D = α R {α = ∧x ((∃ ∧
Df.,
∧
∃x) . xRy)} Df., [D] = β R {β = ∧y ((∃ ∧ ∧
∃y) : xRy . ∨ . yRx)} C = γ R {γ = ((∃
Df.
AZf_lbf qlh lj_lv_ ba wlbo hij_^_e_gbc agZqbfh lhevdh lh]^Z dh]^Z R _klv lh qlh fh`gh [ueh [u gZa\Zlv h^ghjh^guf hlghr_gb_ l_ hlghr_gb_f \ dhlhjhf _keb xRy bf__lf_klhobmhlghkylkydh^ghfmblhfm`_lbim<ijhlb\ghfkemqZ_dZd[ufug_ \u[bjZebobmeb[hxRyeb[hyRx[ueb[u[_kkfuke_ggufbWlhgZ[ex^_gb_\Z`gh\ k\yabkiZjZ^hdkhf;mjZeb-Nhjlb GZhkgh\Zgbbijb\_^zgguohij_^_e_gbcfuihemqZ_f ∃y) . xRy}, L. D‘R = ∧x {(∃ ∧ ∃x) . xRy}, L. [D]‘R = y {(∃ ∃y) : xRy . ∨ . yRx}, L. C‘R = ∧x {(∃
ihke_^g__[m^_l agZqbfh lhevdh lh]^Z dh]^Z R h^ghjh^ghµD‘R¶ qblZ_lky dZdµh[eZklv R’; ‘[D]‘R¶qblZ_lkydZdµdhg\_jkgZyh[eZklvR’; ‘C‘R¶qblZ_lkydZdµihe_R’. >Ze__gZflj_[m_lkyaZibkv^eyhlghr_gbydeZkkZqe_gh\ddhlhjufg_dhlhjucwe_ f_gl ba α bf__l hlghr_gb_ R d deZkkm α kh^_j`Zs_fmky \ h[eZklb R Z lZd`_ aZibkv ^eyhlghr_gbydeZkkZqe_gh\dhlhju_bf_xlhlghr_gb_Rdg_dhlhjhfmwe_f_glmba β, ddeZkkm βkh^_j`Zs_fmky\dhg\_jkghch[eZklbR>ey\lhjhcbagbofumklZgZ\eb\Z _f ∧
Ihwlhfm
∧
∃y) . y∈β . xRy)} R∈ = α β {α = ∧x ((∃
Df.
∃y) . y∈β . xRy}. L. R∈‘β = ∧x {(∃
LZd_kebR_klvhlghr_gb_hlpZdkugmZ β –wlhdeZkk\uimkdgbdh\BlhgZlhR∈‘β[m ^_ldeZkkhfµhlpu\uimkdgbdh\BlhgZ¶ _kebR_klvhlghr_gb_µf_gvr_¶Zβ –wlhdeZkk ijZ\bevguo ^jh[_c nhjfu –2–n ^ey p_euo agZq_gbc n lh R∈‘β [m^_l deZkkhf ^jh[_c f_gvrboq_fg_dhlhjZy^jh[vnhjfu–2–n l_R∈‘β[m^_ldeZkkhfijZ\bevguo^jh[_c >jm]h_\ur_mihfygmlh_hlghr_gb__klv R )∈. <dZq_kl\_Zevl_jgZlb\ghcaZibkbqZklh[he__m^h[ghcfumklZgZ\eb\Z_f ∪
R‘‘β = R∈‘β Df. Hlghkbl_evgh_ ijhba\_^_gb_ ^\mo hlghr_gbc R b S _klv hlghr_gb_ dhlhjh_ bf__l f_klhf_`^mobz\k_]^Zdh]^Zbf__lkywe_f_glmlZdhcqlhbxRybyRzbf_xlf_klh Hlghkbl_evgh_ijhba\_^_gb_h[hagZqZ_lkydZdRSLZd RS = FulZd`_mklZgZ\eb\Z_f
∧∧ ∃y) x z {(∃
. xRy . yRz}
R2 = RR
Df.
Df.
QZklh lj_[mxlky ijhba\_^_gb_ b kmffZ deZkkZ deZkkh\ Hgb hij_^_eyxlky ke_^mx sbfh[jZahf ∃α) . α∈κ . x∈α} s‘κ = ∧x {(∃ ∧ p‘κ = x {α∈κ . ⊃α . x∈α}
Df. Df.
Koh^gufh[jZahf^eyhlghr_gbcfumklZgZ\eb\Z_f . s ‘λ = ∧x ∧y {(∃ ∃R) . R∈λ . xRy}
Df.
. p ‘λ = ∧x ∧y {R∈λ . ⊃R . xRy}
Df.
GZfgm`gZaZibkv^eydeZkkh\qvbf_^bgkl\_ggufwe_f_glhfy\ey_lkyoI_Zghbk ihevam_lιxihwlhfmfu[m^_fbkihevah\Zlvι‘xI_ZghihdZaZe wlhih^qzjdb\ZebNj_ ]_ qlh wlhl deZkkg_evay hlh`^_kl\blv k o Ijbh[uqghf \a]ey^_gZ deZkkug_h[oh^b fhklv lZdh]h jZaebqby hklZzlky aZ]Z^hqghc gh k lhqdb aj_gby \u^\bgmlhc \ur_ hgZ klZgh\blkyhq_\b^ghc FumklZgZ\eb\Z_f ∧
ι = α ∧x {α = ∧y (y = x)}
hlkx^Z
Df.,
L. ι‘x = ∧y (y = x) Df.,
b
L: E! ι ‘α . ⊃ . ι ‘α = ( ∪
∪
x)(x∈α);
l__keb α –wlhdeZkkdhlhjucbf__llhevdhh^bgwe_f_gllhwlbfwe_f_glhfy\ey_lky ∪
ι ‘α . 1
>eydeZkkZdeZkkh\kh^_j`Zsboky\^ZgghfdeZkk_fumklZgZ\eb\Z_f ∧
Cl‘α = β (β ⊂ α) Df. L_i_jvfufh`_fi_j_clbdjZkkfhlj_gbxdZj^bgZevguobhj^bgZevguoqbk_eblh ]hdZdboaZljZ]b\Z_lmq_gb_hlbiZo IXD:J>BG:EVGU?QBKE: DZj^bgZevgh_qbkehdeZkkZ αhij_^_ey_lkydZddeZkk\k_odeZkkh\koh^guok α ^\Z deZkkZ y\eyxlky koh^gufb dh]^Z f_`^m gbfb bf__lky h^gh-h^ghagZqgh_ hlghr_gb_ DeZkkh^gh-h^ghagZqguohlghr_gbch[hagZqZ_lkydZd→bhij_^_ey_lkyke_^mxsbf h[jZahf ∧
1→1 = R {xRy . x/Ry . xRy/ . ⊃x, y, x/, y/ . x = x/ . y = y/}
Df.
Koh^kl\hh[hagZqZ_lkydZdSimbhij_^_ey_lkylZd ∧
∧
Sim = α β {(∃ ∃R) . R∈1→1 . D‘R = α . D‘R = β}
Df.
Lh]^Z Sim ‘α_klvihhij_^_e_gbxdZj^bgZevgh_qbkeh α _]hfu[m^_fh[hagZqZlvdZd Nc‘α ke_^h\Zl_evghfumklZgZ\eb\Z_f →
1
∪
LZdbfh[jZahf ι ‘α_klvlhqlhI_ZghgZau\Z_lια.
→
Nc = Sim
hlkx^Z
Df.,
L. Nc‘α = Sim ‘α. →
DeZkkdZj^bgZevguoqbk_efu[m^_fh[hagZqZlvdZdNC lZdbfh[jZahf NC = Nc‘‘cls
Df.
hij_^_ey_lky dZd deZkk qvbf _^bgkl\_gguf we_f_glhf y\ey_lky gmev-deZkk l_ Λ), ihwlhfm 0 = ι‘Λ
Df.
Hij_^_e_gb_ke_^mxs__ ∧
∃c) : x∈α . ≡x . x = c} Df. 1 = α {(∃ E_]dh^hdZaZlvqlhkh]eZkghhij_^_e_gbxby\eyxlkydZj^bgZevgufbqbkeZfb H^gZdhg_h[oh^bfhhlf_lblvqlhkh]eZkghijb\_^zgguf\ur_hij_^_e_gbyfb \k_^jm]b_dZj^bgZevgu_qbkeZy\eyxlkyg_hij_^_ezggufbkbf\heZfblbiZclsbbf_ xlklhevfgh]hagZq_gbckdhevdhkms_kl\m_llbih\GZqgzfk agZq_gb_aZ\bkblhl agZq_gbyΛZagZq_gb_ΛjZaebqZ_lkykh]eZkghlbimgmev-deZkkhfdhlhjh]hhgy\ey_lky LZdbfh[jZahfkms_kl\m_lklhevdh`_kdhevdhkms_kl\m_llbih\ lh`_kZfh_ijbf_ gy_lkydh\k_f^jm]bfdZj^bgZevgufqbkeZfL_fg_f_g___keb^\ZdeZkkZ αb βhlgh kylkydjZaebqguflbiZffufh`_f]h\hjblvhgbodZdh[bf_xsboh^ghblh`_dZj ^bgZevgh_qbkehbeb qlhh^bg ba gbo bf__l dZj^bgZevgh_ qbkeh[hevr__ q_f ^jm]hc ihkdhevdmh^gh-h^ghagZqgh_hlghr_gb_fh`_lbf_lvf_klhf_`^mwe_f_glZfb αb β^Z `_ lh]^Z dh]^Z α b β hlghkylky d jZaebqguf lbiZf GZijbf_j imklv β [m^_l ι‘‘α l_ deZkkhfqvbfbwe_f_glZfby\eyxlkydeZkkukhklhysb_ba_^bgkl\_ggh]hqe_gZ αLh ]^Z ι‘‘αhlghkblkyd[he__\ukhdhfmlbimq_f αghih^h[gh αihkdhevdmkhhlg_k_ghk αihkj_^kl\hfh^gh-h^ghagZqgh]hhlghr_gbyι. B_jZjobylbih\bf__l\Z`gu_ke_^kl\by\hlghr_gbbkeh`_gbyIj_^iheh`bfmgZk _klvdeZkkba αqe_gh\bdeZkkba βqe_gh\]^_ αb βy\eyxlkydZj^bgZevgufbqbkeZfb fh`_lkemqblvkylZdqlhbokh\_jr_gghg_\hafh`ghh[t_^_gblvqlh[uihemqZlvdeZkk khklhysbcbaqe_gh\ αbbaqe_gh\ βihkdhevdm_kebdeZkkug_hlghkylkydh^ghfmb lhfm`_lbimboeh]bq_kdZykmffZ[_kkfuke_ggZLhevdhlZf]^_jZkkfZljb\Z_fh_qbk ehdeZkkh\dhg_qghfufh`_fmkljZgblvijZdlbq_kdb_ke_^kl\bywlh]h[eZ]h^Zjylhfm nZdlmqlhfu\k_]^Zfh`_fijbf_gblvddeZkkmdhlhjucm\_ebqb\Z_lk\hclbi^hex [hc lj_[m_fhc kl_i_gb [_a baf_g_gby k\h_]h dZj^bgZevgh]h qbkeZ GZijbf_j ijb ex [hf deZkk_ α deZkk ι‘‘α bf__l lh `_ kZfh_ dZj^bgZevgh_ qbkeh gh hlghkblky d lbim b^ms_fmaZ αKe_^h\Zl_evgh^eyex[h]hdhg_qgh]hqbkeZdeZkkh\jZaebqguolbih\fu fh`_fm\_ebqblv\k_bo^hlbiZdhlhjucfufh`_fgZa\ZlvgZbf_gvrbfh[sbffgh `bl_e_f \k_o jZkkfZljb\Z_fuo lbih\ b fh`gh ihdZaZlv qlh wlh fh`_l [ulv k^_eZgh lZdbfkihkh[hfqlhj_amevlbjmxsb_deZkkug_[m^mlbf_lvh[sbowe_f_glh\AZl_ffu fh`_fh[jZah\Zlveh]bq_kdmxkmffm\k_oihemq_gguolZdbfh[jZahfdeZkkh\b_zdZj
^bgZevgh_qbkeh[m^_lZjbnf_lbq_kdhckmffhcdZj^bgZevguoqbk_ebagZqZevguodeZk kh\GhlZf]^_mgZk_klv[_kdhg_qgu_ihke_^h\Zl_evghklbdeZkkh\\hkoh^ysbolbih\ wlhlf_lh^ijbf_gblvg_evayIhwlhcijbqbg_fug_fh`_f^hdZaZlvqlh^he`gu[ulv [_kdhg_qgu_deZkkuB[hij_^iheh`bfqlh[ueh[u\hh[s_lhevdhnbg^b\b^h\]^_n –dhg_qghLh]^Z[ueh[u 2 n deZkkh\bg^b\b^h\ 2 2 deZkkh\deZkkh\bg^b\b^h\bl^LZ dbfh[jZahfdZj^bgZevgh_qbkehqe_gh\dZ`^h]hlbiZ[ueh[udhg_qgh bohlywlbqbk eZij_\hkoh^beb[uex[h_aZ^Zggh_dhg_qgh_qbkehg_[ueh[ukihkh[Zkeh`blvbolZd qlh[uihemqblv[_kdhg_qgh_qbkehKe_^h\Zl_evghgZfg_h[oh^bfZ bih\k_c\b^bfh klb lZd hgh b _klv ZdkbhfZ \ lhf kfuke_ qlhgbh^bg dhg_qguc deZkkbg^b\b^h\ g_ kh^_j`bl\k_bg^b\b^u h^gZdh_kebdlh-lhhl^Zklij_^ihql_gb_lhfmqlhh[s__qbkeh bg^b\b^h\\mgb\_jkmf_jZ\ghkdZ`_flhih-\b^bfhfmg_lZijbhjgh]hkihkh[Z hijh\_j]gmlv_]hfg_gb_ GZ hkgh\Zgbb ij_^eh`_ggh]h \ur_ kihkh[Z jZkkm`^_gby ykgh qlh ^hdljbgZ lbih\ ba[_]Z_l\k_oaZljm^g_gbchlghkbl_evghgZb[hevr_]hdZj^bgZevgh]hqbkeZGZb[hevr__ dZj^bgZevgh_qbkeh_klv\dZ`^hflbi_ gh_]h\k_]^Zij_\hkoh^bldZj^bgZevgh_qbkeh ke_^mxs_]hlbiZihkdhevdm_kebα –dZj^bgZevgh_qbkehh^gh]hlbiZlhdZj^bgZevgh_ qbkehke_^mxs_]hlbiZ_klv 2α dhlhjh_dZdihdZaZeDZglhj\k_]^Z[hevr_q_f αIh kdhevdmg_kms_kl\m_lf_lh^Zkeh`_gbyjZaebqguolbih\fug_fh`_f]h\hjblvhµdZj ^bgZevghfqbke_\k_oh[t_dlh\dZdbo[ulhgb[uehlbih\¶bihwlhfmZ[khexlghgZb [hevr_]hdZj^bgZevgh]hqbkeZg_kms_kl\m_l ?kebijbgbfZ_lkyqlhgbh^bgdhg_qgucdeZkkbg^b\b^h\g_kh^_j`bl\k_obg^b\b ^h\hlkx^Zke_^m_lqlhkms_kl\mxldeZkkubg^b\b^h\bf_xsb_ex[h_dhg_qgh_qbk eh Ke_^h\Zl_evgh \k_ dhg_qgu_ dZj^bgZevgu_ qbkeZ bf_xl f_klh dZd dZj^bgZevgu_ qbkeZbg^b\b^h\ l_ dZd dZj^bgZevgu_ qbkeZ deZkkh\ bg^b\b^h\ Hlkx^Z ke_^m_l qlh kms_kl\m_l deZkk 0 dZj^bgZevguo qbk_e Z bf_ggh deZkk dhg_qguo dZj^bgZevguo qb k_eKe_^h\Zl_evgh 0bf__lf_klhdZddZj^bgZevgh_qbkehdeZkkZdeZkkh\deZkkh\bg ^b\b^h\H[jZamy\k_deZkkudhg_qguodZj^bgZevguoqbk_efugZoh^bfqlh 2ℵ bf__l f_klhdZddZj^bgZevgh_qbkehdeZkkZdeZkkh\deZkkh\deZkkh\bg^b\b^h\ blZdfufh `_fijh^he`Zlvg_hij_^_ezggh^he]hFh`ghlZd`_^hdZaZlvkms_kl\h\Zgb_ n^eydZ `^h]hdhg_qgh]hn ghwlhlj_[m_ljZkkfhlj_gbyhj^bgZeh\ ?keb\^h[Z\hddij_^iheh`_gbxqlhgbh^bgbadhg_qguodeZkkh\g_kh^_j`bl\k_o bg^b\b^h\fuij_^iheZ]Z_ffmevlbiebdZlb\gmxZdkbhfm l_Zdkbhfmqlh^eyaZ^Zg gh]h fgh`_kl\Z \aZbfgh bkdexqZxsbo deZkkh\ gb h^bg ba dhlhjuo g_ y\ey_lky gme_ \uf _klv ih djZcg_c f_j_ h^bg deZkk \dexqZxsbc h^bg we_f_gl ba dZ`^h]h deZkkZ wlh]hfgh`_kl\Z lhfufh`_f^hdZaZlvqlhkms_kl\m_ldeZkkkh^_j`Zsbc 0we_f_g lh\ lZd qlh 0 [m^_l bf_lv f_klh dZd dZj^bgZevgh_ qbkeh bg^b\b^h\ Wlh g_kdhevdh mf_gvrZ_llbi^hdhlhjh]hfu^he`gu^hclbqlh[u^hdZaZlvl_hj_fmhkms_kl\h\Zgbb ^eyex[h]haZ^Zggh]hdZj^bgZevgh]hqbkeZghg_^ZzlgZfdZdhc-eb[hl_hj_fuhkms_ kl\h\ZgbbdhlhjZyjZgvr_bebiha`_g_fh`_l[ulvihemq_gZbgZq_ Fgh]b_we_f_glZjgu_l_hj_fu\dexqZxsb_dZj^bgZevgu_qbkeZlj_[mxlfmevlbi ebdZlb\gmx Zdkbhfm1 G_h[oh^bfh hlf_lblv qlh wlZ ZdkbhfZ wd\b\Ze_glgZ Zdkbhf_ P_jf_eh2bke_^h\Zl_evgh^hims_gbxqlhdZ`^ucdeZkkfh`_l[ulv\iheg_mihjy^h n
0
Kj.: qZklv III fh_c klZlvb ‘On some Difficulties in the Theory of Transfinite Numbers and Order Types’, Proc. London Math. Soc. Ser. II, Vol. IV, Part I. 2 H[Zdkbhf_P_jf_ehbh^hdZaZl_evkl\_lh]hqlhwlZZdkbhfZ\e_qzlfmevlbiebdZlb\gmxZdkbhfmkf ij_^u^msmxkghkdmH[jZlguc\u\h^\u]ey^bllZd H[hagZqbfdZdProd‘kfmevlbiebdZlb\gucdeZkk kjZkkfhljbf 1
q_g1Wlbwd\b\Ze_glgu_ij_^ihkuedbih-\b^bfhfm^hdZaZlvg_\hafh`ghg_kfhljygZ lhqlhfmevlbiebdZlb\gZyZdkbhfZ\u]ey^bl^hklZlhqghijZ\^hih^h[ghc<hlkmlkl\bb ^hdZaZl_evkl\Z\b^bfhemqr_g_ijbgbfZlvfmevlbiebdZlb\gmxZdkbhfmdZd^hims_ gb_ghmklZgZ\eb\Zlv_zdZdmkeh\b_\dZ`^hfkemqZ_\dhlhjhfhgZbkihevam_lky
XHJ>BG:EVGU?QBKE: Hj^bgZevgh_qbkeh_klvdeZkkhj^bgZevghkoh^guo\iheg_mihjy^hq_gguojy^h\l_ hlghr_gbch[jZamxsbolZdb_jy^uHj^bgZevgh_koh^kl\hbebih^h[b_hij_^_ey_lky ke_^mxsbfh[jZahf ∧
∧
∪
∃S) . S∈1→1 . [D]‘S = C‘Q . P = SQ S } Smor = P Q {(∃
Df.,
]^_µSmor¶_klvkhdjZs_gb_^eyµkoh^guhj^bgZevgh¶ DeZkkhlghr_gbcjy^Zdhlhju_fu[m^_fgZau\ZlvµSer¶hij_^_ey_lkylZd ∧
→
←
Ser = P {xPy . ⊃x, y . ∼ (x = y) : xPy . yPz . ⊃x, y, z . xPz : x∈ C‘P . ⊃x . P ‘x ∪ ι‘x ∪ P ‘x = C‘P}Df. L__kebqblZlvJdZdµij_^r_kl\m_l¶lhhlghr_gb_y\ey_lkyhlghr_gb_fjy^Z_keb g_lgbh^gh]hwe_f_glZij_^r_kl\mxs_]hkZfhfmk_[_ ij_^r_kl\_ggbdij_^r_ kl\_ggbdZ_klvij_^r_kl\_ggbd _kebo_klvdZdhc-lhqe_giheyhlghr_gbylhij_^ r_kl\_ggbdb o \f_kl_ k o \ kh\hdmighklb k _]h ij_^r_kl\_ggbdZfb h[jZamxl \kz ihe_ hlghr_gby
∪
Ω = P {P∈ Ser : α ⊂ C‘P . ∃!α . ⊃α . ∃!(α – P ‘‘α)}
Df.;
l_ P ihjh`^Z_l \iheg_ mihjy^hq_ggu_ jy^u _keb J _klv hlghr_gb_ jy^Z b ex[hc deZkk αkh^_j`Zsbcky\ihe_Jbg_y\eyxsbckygme_\ufbf__li_j\ucqe_g Hlf_ lbfqlh P ‘‘αkmlvqe_gu\oh^ysb_ihke_g_dhlhjh]hqe_gZα). ?kebdZdNo‘Ph[hagZqblvhj^bgZevgh_qbkeh\iheg_mihjy^hq_ggh]hhlghr_gbyJZ dZdNOdeZkkhj^bgZevguoqbk_elhfuihemqbf ∪
∧
∧
→
No = α P {P∈Ω . α = Smor ‘P} Df., NO = No‘‘Ω. ∧
bij_^iheh`bfqlh
∃x) . x∈β . D‘R = ι‘β . [D]‘R = ι‘x} Z‘β = R {(∃ ∧ ∧
Df.,
γ∈ Prod‘Z‘‘cl‘a . R = ξ x {(∃ ∃S) . S∈γ . ξSx}. Lh]^ZR –wlhkhhl\_lkl\b_P_jf_ehKe_^h\Zl_evgh_kebProd‘Z‘‘cl‘ag_y\ey_lkygme_\uflh^eyZ kms_kl\m_lihdjZcg_cf_j_h^ghkhhl\_lkl\b_P_jf_eh 1 Kf.: Zermelo, ‘Beweis, dass jede Menge wohlgeordnet werden kann’. Math. Annalen, Vol. LIX, C.514-16.
Bahij_^_e_gbyNofuihemqZ_f L: P∈ Ω . ⊃ . No‘P = Smor ‘P , L: ∼(P∈ Ω) . ⊃ . ∼E!No‘P. →
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