Ramsey and the Vienna Circle 08

J.W. Degen

Logical Problems Suggested by Logicism

The mathematics of logic is difficult, the logic of mathematics is even more difficult.

1. Introduction Let us call the Logicist Thesis, or the Thesis of Logicism, or simply Logicism the thesis that [LT] Pure Mathematics is part of Logic. The founding fathers of Logicism are Frege and Russell. Roughly, Frege maintained that (at least) higher-order arithmetic is part of logic, but definitively not geometry, whereas for Russell even all of pure mathematics was to be part of logic. Unfortunately, Frege’s system GGA (Grundgesetze der Arithmetik, 1893, 1903) [6] by means of which he wanted to prove his version of [LT] was shown to be inconsistent by Russell. However, had GGA been consistent it would have proved, provided it is logic, a much stronger version of [LT] than Frege envisaged since GGA contains all of set theory despite the more modest title Grundgesetze der Arithmetik.1 Russell’s manifesto of his Logicism is to be found in his Principles of Mathematics of 1903; it is firmly repeated in the second edition of 1938 [11]2. The formal implementation followed 1908 in Russell’s Mathematical logic as based on the theory of types [12], and then in the Principia Mathematica written with Whitehead, published 1910–13. The second edition of PM of 1927 [14] seems to be still in print. The sentence [LT] as stated above contains three undefined phrases: (1) Pure Mathematics (2) is part of (3) Logic Furthermore, even if these three notions are defined in some way or other, there remains the following ambiguity in [LT]: nonuniform[LT]: For every (sharply delineated) piece M of Pure Mathematics there exists a logic LM such that M is part of LM .

123 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 123–138. © 2006 Springer. Printed in the Netherlands.

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uniform[LT]: There is one universal logic Luniv such that all of Pure Mathematics is part of Luniv . Finally, we have the following weak version of the logicist thesis. weak[LT]: The (or some) main part of Pure Mathematics is part of some Logic. The cautious distinctions just given are necessitated by the mathematical and logical experience between 1903 and 2003. It is possible to refute even the version weak[LT]. For instance, let us say that Logic is just first-order logic, and that each main part of pure mathematics must contain some nontrivial arithmetic. Then weak[LT] is false. Also, even if we admit pure classical type theory CT , i.e. P M \ inf inity as a logic, weak[LT] will become false. On the other hand, if we admit ZF C as Logic, and are not squeamish about the vast incompleteness of ZF C, then we may even argue for uniform[LT]. Considering as the main part of Pure Mathematics ordinary mathematics as known by Russell, namely classical analysis, algebra and certain parts of Cantor’s set theory (below ℵω ) and admitting P M as logic, then at least weak[LT] can be vindicated.

2. Some Preparatory Clarifications In my talk I do not want to refute any of these logicist theses, not even the strongest among them. Rather, I will prove (a version of) weak[LT]. Furthermore, I claim that my proof is non-trivial and will yield new information and logical (or mathematical) problems about the logical (or mathematical) status of CT , P M , extensions and variants thereof. I must admit that a philosophically satisfactory analysis of Logicism should dwell more carefully on big questions such as: What is mathematics? What is Logic? I decide these questions by fiat in order not to impair my message by difficulties extraneous to its (rather precisely statable) mathematicallogical content.3 2.1 Rich Model-theoretic Logics For our purpose, let us define a model-theoretic logic L to be a triple (M, |=, L). Here M is a class of models, L is a language4 conceived of as a class of sentences; and for M ∈ M and ϕ ∈ L we mean by M |= ϕ that M makes ϕ true. By V alid(L) we understand the valid sentences of L, i.e. those sentences which are true in all models M ∈ M. We call a (model-theoretic) logic rich if M is large, i.e. a big set or a proper class of pairwise non-equivalent5 models. If L is rich then V alid(L) captures the intuitive notion of a universally (or logically) true sentence, i.e. a sentence true under all possibilities or true in all possible worlds. Of course, if M consists of just one or two models, then V alid(L) is far from being a set of universally true sentences, in the intuitively required sense.6

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Let now L be a rich logic, and P a part of pure mathematics. We will say that ∗ L (semantically) captures P if there is a syntactic interpretation (−) such that (I) For all results π of P : π ∗ ∈ V alid(L). Thus far, we have not said anything about the way P is presented; nor what it means that π is a result of P . Indeed, we have not said what P really is. It may be a structure or a theory, or an activity or whatnot. However, it must be something definite if we want to prove something about it. Our discussion hitherto is connected with Logicism by the stipulation that the interpretation π ∗ is to be a universally valid sentence of a rich logic. Nothing is stipulated about the genuine or intrinsic logicality of the notions representable by the language L of the model-theoretic logic L. In spite of this generality, several possibilities are already ruled out, e.g., the case that π ∗ is a set-theoretic sentence (of the first-order standard language {∈} of set theory) which is true just in the model (Vω+ω , ∈). But it does not rule out the related case where the model class considered is Mzermelo = {(Vα , ∈) : α a limit ordinal ≥ ω + ω}. Definition (I) gives a semantic version of weak[LT] with respect to the rich model-theoretic logic L, and the chosen part P of pure mathematics. Now, if P is ordinary pure mathematics formulated in set-theoretic terms, and we take the rich model-theoretic logic Lzermelo = (Mzermelo , |=, {∈}), then Lzermelo captures P (semantically). Neither Frege nor Russell had such a purely semantic version of Logicism in mind, although there exists a precise one, as just explained. Nevertheless, something like our semantic version of Logicism was surely implied by their logicisms. Moreover, Frege and Russell had (also) some system of proofs in mind, and – being of the highest importance for their project – several lists of so-called logical definitions of mathematical concepts. However, the semantic version of Logicism has, as a foundational standpoint, no great a priori plausibility, and the possible fruits of a realization of this version are rather dubious. For let us take as an example of pure mathematics the (or a) second-order theory T of the real numbers qua completely ordered field (this is really third-order arithmetic in so far as real numbers can be modelled as sets of natural numbers). Then, why should every theorem ϕ of T translate into a sentence ϕ∗ ∈ V alid(L) for some rich model-theoretic logic L? That is to say: why should all or at least many sentences which are true in some individual structure translate into sentences which are true in all structures of a certain kind? Neither Frege nor Russell had the conceptual tools even to formulate this question. But we can see already from these considerations that there is a big lack of motivation in the very idea of Logicism, at least when formulated semantically; for, why should a sentence which is true in few mathematical structures, or just in the field of real numbers, not belong to pure mathematics? Does truth in all structures endow a sentence with a dignity over and beyond those sentences which are true in just one structure?

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2.2 Logical proofs Besides the rich logic L we are now going to introduce proofs into our picture since, as just mentioned, both Frege and Russell explicitly envisaged proofs as part of their logicist programme. Suppose that Σ is a system of proofs which is correct with respect to L, i.e. if Σ  ϕ, then ϕ ∈ V alid(L). Then we will say that Σ captures (syntactically) the piece P of mathematics if (II) For all results π of P : Σ  π ∗ . Of course, (II) implies (I), but not the other way around unless Σ is (semantically) complete with respect to (the model class of) L. Version (II) commands the most interest when the proofs of the proof system Σ are logical proofs. But what is a logical proof? An unobjectionable definition would run like this. Let a rich logic L be given. A proof is a sort of tree whose nodes are sentences from the language L connected by applications of inference rules. Such a proof is logical (with respect to L) if its leaves are members of V alid(L) and the inferences preserve membership in V alid(L). Note that nothing in our definitions presupposes that the sentences of the rich logic L are finite symbolic configurations, or that the proofs in Σ are finite trees. Moreover, we do not assume that the models in L are finite. Why should we? 3. The Argument Now I will present the promised proof; it will use a certain system Σℵ1 of logical proofs, and two associated rich logics. In order to prove (mathematically, or logically) my claim that Σℵ1 syntactically (and therefore semantically) captures a part P of mathematics, this P must be made precise. Although it may seem, prima facie, both logical and historical nonsense, we set P := P M , that is, unramified Principia Mathematica with a full comprehension schema and an axiom of infinity. The perfectly exact definition of P M will be given presently. I have promised a nontrivial proof of weak[LT]. If P M is a logic (and I do not deny this), then we have a proof of weak[LT] via P M  π =⇒ P M  π ∗ ∗ with (−) the identity function. Certainly, this proof is trivial. Regardless of whether P M is a logic or not, Σℵ1 will be logical in a higher degree than P M , and that in a precise sense of logical; moreover, Σℵ1 will turn out to be a natural and systematic strengthening of P M . That P M , in turn, captures a large part of mathematics is well known; it captures more mathematics than anyone of us will ever learn. Disregarding incompletenesses of G¨odelian 1931-type and purely set-theoretic questions like hypotheses about the continuum, P M is “practically complete" for ordinary mathematics (perhaps when enlarged by forms of the axiom of choice). Thus, if P M as it stands is a logic, Russell and Whitehead have proved Logicism at least in the version weak[LT].

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The announced system Σℵ1 is just one in an infinite hierarchy of type logics all of which have a very simple definition. This hierarchy, along with two related hierarchies concerning forms of AC, are fully presented and investigated in my paper [3]. First, the types. Let κ be ℵ0 , ℵ1 , ℵ2 , . . .7 Then the κ-types are defined as follows (1) 0 is a κ-type. (2) if α < κ, and if (τξ )ξ<α is an α-sequence of κ-types, then [(τξ )ξ<α ] is a κ-type. If κ = ℵ0 , then we have just the structure of all finite types, which underlies PM. Formulae and terms of the various  κ-types are built up as expected. We note that we use negation ¬, conjunctions , quantifications ∀, and relation-abstraction λ, the last three of any length α < κ. Besides the notions just listed, we have for α < κ a relation of α-ary typed predication8 denoted simply by brackets, −(−−−), thus τ

T [(τξ )ξ<α ] (T0τ0 , . . . , Tξ ξ , . . .) . This defines the type-theoretic language Lκ . In the case κ = ℵ1 , an expression of Lκ may be infinite, but its length α is a countable ordinal, i.e. α ∈ ℵ1 . The language Lκ may contain for certain types τ several nonlogical constants of type τ . If not, Lκ may be called the pure language. The proof system Σκ is then a sequent calculus consisting of the following: Axioms of the formϕ =⇒ ϕ. Weak structural rules. Rules for the left and right introduction of ¬, , ∀ and λ (together with −(− − −)). As a special feature, these rules allow the introduction of arbitrarily many signs of the same sort at once (so-called block-inferences). No cut rule is assumed.9 We could assume, but do not, an extensionality rule. As a first example we give a special case of the rule that introduces λ in the succedent: τ

[1]( =⇒ λ)

Φ =⇒ Ψ, F[(Tξ ξ )ξ<α ] τ

τ

τ

Φ =⇒ Ψ, λ(Xξ ξ )ξ<α .F[(Xξ ξ )ξ<α ]((Tξ ξ )ξ<α )

Here [1] in [1]( =⇒ λ) signifies that just one λ is introduced. As we have said, we can introduce arbitrarily many λ’s into the succedent (and also into the antecedent), since we have block inferences. A second example is the following rule with infinitely many premisses. 

 Φ =⇒ Ψ, ϕi i<ω  Φ =⇒ Ψ, (ϕi )i<ω  If we want to introduce more than one at once into the succedent, the premisses will become more complicated. [1]( =⇒

)

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There are two rich model-theoretic logics associated with Σκ : (1) StdLκ = (Mκ , |=, Lκ ). Here, an element M from Mκ consists of a domain M τ for each κ-type τ . M 0 is simply any non-empty set of things (socalled individuals). M [(τξ )ξ<α ] is the set of all α-ary relations whose arguments are from the M τξ ’s. These are the standard (or full) models. (2) GenLκ = (Hκ , |=, Lκ ) is like StdLκ but with a sort of Henkinean general models instead of the standard models Mκ . In an H ∈ Hκ a domain H [(τξ )ξ<α ] ∈ H may be only a definitionally closed subset of the set of all pertinent relations. Then we can prove (see [3]) (a) Σκ + (cut) is correct with respect to GenLκ (and hence also with respect to StdLκ ); and (b) Σκ [without (cut)!] is complete with respect to GenLκ . But even Σκ +(cut) is incomplete with respect to StdLκ .10 From (a) and (b) we have cut elimination for all Σκ . This was known as Takeuti’s Conjecture in the case of Σℵ0 and was proved by Prawitz and Takahashi. The case of Σℵ0 is excellently presented in Sch¨utte’s Proof Theory [13]. Now for our precise version of P M This is Σℵ0 + (cut) enlarged by the following “nonlogical axiom” concerning a binary relation constant R[00] (inf inity) : =⇒ ∀x0 ¬R[00] (x0 , x0 ) ∧ R[00] trans ∧ ∀x0 ∃y 0 R[00] (x0 , y 0 ) Obviously, the sequent (inf inity) has no finite model. Using the sequent (inf inity) we can prove in (our precise) P M that each Frege-Russell natural number11 is non-empty, which is the form of the axiom of infinity adopted in Principia Mathematica. Note that the sequent (inf inity) does belong neither to V alid(StdLℵ0 ) nor to V alid(GenLℵ0 ). In this precise sense, P M is not a “pure" logic. Remark. The version of P M just defined is consistent. If this were not the case, then in Σℵ0 + (cut) we could prove the sequent (††) ∀x0 ¬R[00] (x0 , x0 ), R[00] is transitive, ∀x0 ∃y 0 R[00] (x0 , y 0 ) =⇒ Then Σℵ0 would also prove (††) by a cut-free proof π. Since (††) contains only first-order quantifiers, π would also be a cut-free proof of (††) in first-order logic. But this supposition can be refuted in primitive recursive arithmetic. This consistency proof (observed by Takeuti already before the solution of his Conjecture) shows the metamathematical strength of the cut elimination property for Σℵ0 .12 The consistency problem for P M was one of the many logical problems suggested by Logicism. We think this problem has by now been solved, although not constructively, i.e. not quite in the sense of Hilbert’s programme. ∗

Next we show that there is an interpretation (−) such that for all sequents S

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If P M  S, then Σℵ1  S ∗ . We further assume that Σℵ1 is based upon the pure language, i.e. upon Lℵ1 without any nonlogical constant. What we have to do is to find a translation of the [τ,τ ] term R[00] into a term R∗ of pure Lℵ1 for some ℵ1 -type τ such that the interpreted ∗ sequent (inf inity) becomes provable in Σℵ1 . There are several options. We develop one such possibility, not necessarily the simplest, in detail in order to show the expressive power of Lℵ1 . The type superscripts are omitted if they can easily be restored. And of course, several obvious abbreviations are used. In particular, (Leibniz) equality is defined by T1σ = T2σ := ∀X [σ] (X(T1 ) → X(T2 )) for  the type we write the conjunction (ϕξ )ξ<α σ.  If (ϕξ )ξ<α is an α-sequence of formulae  as ξ<α ϕξ . Similarly for disjunctions . Henceforth, let τ be the ω-long type [[ ][ ][ ] . . .]. Observe that [ ] is a type by our definition; it is the type of 0-ary relations, i.e. of propositions. Define F := ∀x[ ] x[ ] and T := ∃x[ ] x[ ] . These propositions are refutable and provable [] [] in Σℵ1 , respectively. Next define SEQ[(Ai )i<ω ] as saying that (Ai )i<ω is an  [] F, T-sequence (of length ω). So we have SEQ[(Ai )i<ω ] : ⇐⇒ i<ω (Ai = T ∨ Ai = F) Next we order these F, T-sequences lexicographically, where F is taken to be smaller than T. That is, we define LEX[(A  i )i<ω , (Bi )i<ω ] : ⇐⇒ SEQ[(Ai )i<ω ] ∧ SEQ[(Bi )i<ω ] ∧  ( i<ω k≤i Ak = Bk ∧ Ai+1 = F ∧ Bi+1 = T) . Then we define [] SEQun[B τ ] : ⇐⇒ ∃(Xi )i<ω (B((Xi )i<ω ) ∧ SEQ[(Xi )i<ω ]) ∧ ∀(Xi )i<ω ∀(Yi )i<ω (SEQ[(Xi )i<ω ] ∧ SEQ[(Yi )i<ω ] ∧ B((Xi )i<ω ) ∧ B((Yi )i<ω ) → ∀Z τ (Z((Xi )i<ω ) → Z((Yi )i<ω ))) . The universe for the interpretation is U [τ ] := λX τ SEQun[X τ ]. [τ τ ] Finally, R∗ (Aτ , B τ ) says that the F, T-sequence in A is lexicographically smaller than the F, T-sequence in B. So we have the formal definition [τ τ ] := λX, Y (U(X)∧U(Y ) ∧ ∃(xi )i<ω ∃(yi )i<ω (X((xi )i<ω )∧Y ((yi )i<ω )∧ R∗ LEX[(xi )i<ω , (yi )i<ω ])). [τ τ ] Then (inf inity)∗ is (inf inity) with R[00] replaced by R∗ and the quantifiers relativized to U [τ ] . This interpretation (−)∗ can be extended to all of PM by translating the type 0 to the type τ throughout, and inductively any ℵ0 -type σ to the corresponding ℵ1 -type σ ∗ . We can rephrase the result just proven as follows: [τ τ ] The pair (U [τ ] , R∗ ) constitutes an inner model of Σℵ1 + (cut) + (inf inity) within the system Σℵ1 +(cut) and hence within Σℵ1 , by the cut-elimination theorem. Since Σℵ1 is trivially consistent, Σℵ1 + (cut) + (inf inity) and a fortiori P M is consistent; this is another, though unnecessarily complicated, consistency proof for PM.

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Remark. The axiom of choice plays no significant role in our considerations. If necessary we assume tacitly the presence of enough choice in our systems. See [3] for extensions of the Σκ ’s by Hilbertian choice terms together with appropriate rules. Thus, there is a logicist justification also of the axiom of choice. 3.1 Summary of this chapter Let us summarize what we have achieved by our proof of weak[LT] which was, I hope, not quite trivial. We have formulated a hierarchy Σκ , κ = ℵ0 , ℵ1 , . . ., of type logics. These are logics in the following sense: (1) They are complete and correct with respect to rich model-theoretic logic whose model class consists of certain Henkinean models. (2) The axioms and rules of each Σκ are logical in that the former are trivial (i.e. obviously tautological) and the latter consists only of introduction rules for  the logical signs ¬, , ∀, λ and typed predication −(− − −). (3) Then we showed how to interpret P M in Σℵ1 . The hard part is (1); in contradistinction to this, (2) and (3) may be called observations. So we can in Σℵ1 prove in a purely logical way that there are infinitely many primes, the Heine-Borel Theorem, the Cantor-Bendixson Theorem, and so on.13But what is this good for? Kreisel would ask: what do we know more about a true mathematical sentence ϕ when we know that ϕ is provable in Σℵ1 ? I am unable to give a satisfactory answer. I think that even the founding fathers of Logicism, Frege and Russell, had no answer to the natural and obvious question: What do we know more about a mathematical truth if we have a purely logical proof of it? Neither in the Grundgesetze der Arithmetik nor in the Principia Mathematica do we find proofs of deeper theorems from mainstream mathematics.14 Of course, Russell and Whitehead were well in the position to include a P M -proof P, say, of the Prime Number Theorem. But then, what would they have done with this purely logical proof P? Would they have unravelled P in order to gain more information about the Prime Number Theorem? Let me add that if we take P M itself as the logic, then Logicism with respect to P M can be merely a Properly Partial Logicism, since Zermelo’s set theory Z cannot be interpreted in P M , and Z is universally regarded as mathematics.15 This non-interpretability follows from the fact that the consistency of P M can be proven in Z; then simply cite G¨odel’s second incompleteness theorem. But there must be a proof of this non-interpretability that avoids G¨odel’s theorem. Find an interesting one!

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4. Reflections on Ramsey 1925 In this section, I will confine myself to a few remarks about Ramsey’s famous paper The foundations of mathematics of 1925 in [10]. First a quotation. “It might be possible to sacrifice infinite well-ordered series to logical rigour, but the theory of real numbers is an integral part of ordinary mathematics, and can hardly be the object of a reasonable doubt. We are therefore justified in supposing that some logical axiom which is true will justify it. The axiom required may be more restricted than the axiom of reducibility, but, if so, it remains to be discovered.” p. XLV of the first volume of the second edition of Principia Mathematica [14] (1927).16 Did Ramsey in his 1925 paper discover such a substitute for the axiom of reducibility? As the quotation shows, Russell did not acknowledge such a discovery. Although Ramsey is mentioned in a footnote for his “valuable criticisms and suggestions", Ramsey’s 1925 paper is not mentioned in any of the three volumes of [14] as far as I could verify. However, as we shall see, Ramsey claims in his 1925 paper to have discovered a substitute for the Axiom of Reducibility. Ramsey apparently never doubted that the mathematical strength of Principia Mathematica is sufficient for ordinary pure mathematics. His main criticism, as is well known, focussed on a version P M † with a certain ramification regime together with the anticlimactic axiom of reducibility. Recall that Russell introduced orders besides types to be able to keep track of the definitional complexity of what he called propositional functions. Thus if T ≡ T [τ1 ...τk ] is a term (for a propositional function) of the indicated type, T may contain quantifiers both of types lower and higher than any of the types τi . Depending on the occurrences of higher-type quantifiers, the term T will get a certain correspondingly high order in addition to its type [τ1 . . . τk ]. One says that the orders ramify the types. Russell used only natural numbers as orders, but for the purpose of fine-tuning one may also use transfinite ordinals. The ramification of types has to be mirrored in the structure of the axioms and/or inference rules: suppose that T has order 100, then the bound variable in the conclusion of a ( =⇒ ∃) [equivalently of a (∀ =⇒ )] applied to T must have also order 100 (or higher), in addition to its type [τ1 . . . τk ]. As a consequence of this ramification regime, several theorems of classical mathematics cannot be proven. A hackneyed example is the least upper bound principle in analysis. Also, despite Russell’s hope to prove the induction principle for natural numbers within the ramification regime, this cannot be done, as was finally shown by J. Myhill in [9] (1974). Already in 1944 G¨odel expressed some doubts; see [7]. The axiom of reducibility (AR) was introduced precisely in order to regain such standard theorems of classical mathematics as the least upper bound principle. (AR) is really an axiom schema; it postulates that every term is equivalent to a term of order 0. In terms of inference rules, this can simply be done by allowing the bound variable in the conclusion of a ( =⇒ ∃)-inference to be of order 0.

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We have just described P M † , i.e. the system which Ramsey attacks in his paper. As explained, (AR) abolishes the ramification regime together with its orders – so that we may reasonably ask what the orders were good for to begin with. 4.1 Ramsey’s criticism of the axiom of reducibility (AR) In this criticism Ramsey makes two main points: (1) (AR) goes far beyond logic; we read pp. 162–3 the following: [the (AR) is] “a genuine proposition, whose truth or falsity is a matter of brute fact, not of logic. It is, therefore, not a tautology in any sense, and its introduction into mathematics is inexcusable.” And on p. 180: “For as I can neither accept the Axiom of Reducibility nor reject ordinary analysis, I cannot believe in a theory which presents me with no third possibility.” These are hard words. (2) The introduction of the ramification regime was motivated by the avoidance of certain paradoxes as the Liar (or more generally, by the dubious vicious-circle principle).17 Ramsey argues that such (linguistic or epistemological) paradoxes do not concern mathematics proper. The other more mathematical paradoxes, as Russell’s own, are disposed of just by introducing types, without refinement by orders. Ramsey also dispels the fear of vicious circles by an example which has become famous afterwards in this context: “... just as we may refer to a man as the tallest in a group, thus identifying him by means of a totality of which he is himself a member without there being any vicious circle.” (p. 192) 4.2 Ramsey’s predicative propositional functions Ramsey’s decisive transformation of P M † (with ramification regime, (AR) and axiom of infinity) into a mathematically equivalent (!)18 theory is based upon a new notion of predicative (propositional) functions contained in Chapter III (pp. 183– 200). This chapter is rich in detail but sometimes difficult to understand. One of the main ideas of Ramsey’s notion of predicativity consists in the real or merely posited elimination of quantifiers in favour of (possibly) infinite conjunctions. So we read on p. 190: “A predicative function of individuals is one which is any truth-function of arguments which, whether finite or infinite in number, are all either atomic functions of individuals or propositions.” And on p. 191: “Now consider the function of x, (ϕ).f (ϕˆ z , x). Is this a predicative function? It is the logical product ... for different ϕ’s ...” Then Ramsey concludes that the function in question is predicative too since he assumes, I think correctly, that the predicative propositional functions are closed under even infinite truth-functions. Ramsey does not give a formal, not even a semi-quasi-formal, system based on the sketched infinitary notion of predicative functions; the time was not yet ripe for this. A direct implementation would lead to an infinitary system like Σℵ1 or a

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predicative subsystem thereof. But Ramsey primarily wanted to give a justification of a new form of P M whose language is finitary. Thus he seems to have used his detour through infinite expressions, notably infinite truth-functions, only to show that quantifiers do not lead out of the realm of his predicative functions. Ramsey says that he does not need (AR). But what does he assume instead? What was, after all, Ramsey’s substitute for the Axiom of Reducibility? Although he does not say so explicitly, he assumes the full comprehension principles in an unramified language, which is virtually our language Lℵ0 ; the principle thus reads (comp)

∃Y [τ1 ...τk ] ∀X1τ1 . . . Xkτk (Y (X1 . . . Xk ) ↔ F[X1 . . . Xk ]) .

where F is any formula, which may contain arbitrarily high quantifiers. Such a form of comprehension is nowadays called impredicative, despite Ramsey’s different nomenclature. In our systems, full comprehension (comp) is deducible by the following unrestricted rules: ( =⇒ ∃), (λ =⇒ ), ( =⇒ λ), ( =⇒ ∀) and some propositional inferences. It is easy to show that (AR) in a ramified context is equivalent to (comp) in an unramified context. In this sense, Ramsey has not replaced the axiom of reducibility by an essentially different axiom. This he himself concedes on p. 207. See [10] p. 207. Since Ramsey assumes also an axiom of infinity, his improved version of P M † is just our P M discussed above. Moreover, P M † and our P M are mutually interpretable. 4.3 Ramsey on the axioms of infinity and choice In the final chapter of his paper, Ramsey discusses the Multiplicative Axiom, i.e. the Axiom of Choice, and the Axiom of Infinity. He thinks “the introduction of these two axioms is not so grave as that of the Axiom of Reducibilty, because they are not in themselves such objectionable assumptions,...” p. 207. One is tempted to say: Quite the contrary. For, if logicality (or tautologicality) is to be preserved, then the axiom of infinity presents an obstacle. Recall that (inf inity) ∈ V alid(StdLℵ0 ). Ramsey claims that the Multiplicative Axiom is, under his interpretation, a tautology. His arguments for that are hard to follow; but they contain the interesting conjecture that the Multiplicative Axiom cannot be proved in P M . (The difficulty of the independence proof depends on the form of the axiom of infinity.) We have not given a logicist justification of the Axiom of Choice in the present talk; but we may mention in passing that we have constructed cut-free systems εΣκ (resp. τ Σκ ) in which axioms of choice can be proven via logical rules for Hilbertian ε-terms (resp. τ -terms). See [3]. I now quote the final passage of Ramsey’s paper in full: “Similarly the Axiom of Infinity in the logic of the whole world, if it is a tautology, cannot be proved, but must be taken as a primitive proposition. And this is the course which we must adopt, unless we prefer the view that all analysis is self-contradictory

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and meaningless. We do not have to assume that any particular set of things, e.g. atoms, is infinite, but merely there is some infinite type which we can take to be the type of individuals.” p. 212. This is heavy going. However, if Ramsey had implemented his infinitary intuitions in a system like Σℵ1 , he would have recognized the Axiom of Infinity as an infinitary tautology, i.e. as a sentence in V alid(StdLℵ1 ) and would have proved it by a proof similar to ours. 4.4 What we have omitted We have not commented upon Ramsey’s dependence on Wittgensteinian ideas and insights. The main point here is the idea of infinitely long propositional tautologies; this idea indeed occurs in the Tractatus19. But Wittgenstein himself does not make much of the infinitary character of his tautologies, he certainly does not use them to correct Principia Mathematica. It is unknown to me what Wittgenstein did remark on Ramsey’s paper, especially on the use of infinitely long tautologies. However, G¨odel made some pertinent critical remarks in [7]. Ramsey’s criticism of (AR) is anticipated in the Tractatus; see 6.1232 and 6.1233. Let us point out that Wittgenstein makes a very haughty but totally mistaken remark on the axiom of infinity, 5.535 “Damit erledigen sich auch alle Probleme, die an solche Scheinsa¨ tze geknu¨ pft waren ... Das, was das Axiom of infinity sagen soll, wu¨ rde sich in der Sprache dadurch ausdru¨ cken, dass es unendlich viele Namen mit verschiedener Bedeutung ga¨ be.” Now, let us extend the system Σℵ0 by the following set of axioms =⇒ ¬ci = cj f or i = j, the ci s being constants of type 0 . Then no axiom of infinity can be proved in the extended system. For this system is locally finite in the sense that each single theorem of it has a finite model. 5. Conclusion and Open Logical Problems Though Ramsey’s invective against (AR) is full of brillant passages it is not sufficiently cogent to justify the “hard words" I have adduced. I have mentioned above in a footnote that Ramsey himself stated that his improved version of Principia Mathematica differs only in the meaning attached to it. For my part, I remain unconvinced about the alledged viciousness of (AR). For, to begin with, ramifications by themselves are both logically and mathematically as interesting as the types. This was borne out by many subsequent mathematical and logical investigations. For instance, G¨odel’s constructible sets are a generalization of Russell’s ramified type structure. Another example is the proof theory of Ramified Analysis, see, e.g., Sch¨utte’s book [13].

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Finally, I do not find anything wrong in introducing Axioms of Reducibility into a given ramified framework. I intentionally used the plural form Axioms; for, one does not need to reduce every order to the order 0, more modest reductions are possible and sensible. It is not clear why any Axiom of Reducibility should be contradictory to Logicism. We close by stating some logical problems suggested by logicism, as announced in the title of our talk. (I) Define and investigate ramified versions of Σℵ1 . Are there (interesting) ramified versions of Σℵ1 (or of Σℵ2 ) which embed all of P M ? (II) It can be shown that Σℵ1 proves finitary sequents, i.e. sequents belonging to Lℵ0 , which are not provable in P M . Nevertheless, such sequents are valid with respect to standard models. This means, that standard validity for finitary sequents is better approximated by Σℵ1 than by Σℵ0 . Next we can use Σℵ2 for a still better approximation, and so on. This phenomenon calls for thorough investigation. (III) In Σℵ1 we can get complete first-order arithmetic. In which system Σκ can we get complete second-order arithmetic? (IV) The language of Σℵ1 should be made more effective, e.g., by using definability in admissible sets. (V) There are many sensible subsystems of Σℵ1 . For instance, to embed P M we do not need all ℵ1 -types as shown by our embedding described above. A second example is Σ 12 ℵ1 . This is the system with only ℵ0 -types but with countably long  ’s and ∀’s. Prove a semantic completeness theorem for Σ 12 ℵ1 ; and perhaps a cutelimination theorem. Since Σ 12 ℵ1 remains consistent if we stipulate that every type be finite, P M cannot be interpreted in Σ 12 ℵ1 . But we can axiomatically enlarge Σ 12 ℵ1 by adding the sequent (inf inity). Then we can deduce complete first-order arithmetic. For that purpose, we can also drop infinitely long ∀’s. (VI) What will happen if we remove from Σℵ1 all quantifiers? It seems that Ramsey (and perhaps also Wittgenstein) at some time had such a system in mind. (VII) Finally, one should develop a methodology for the construction and presentation of infinitary proofs. These all are logical problems which can be raised about Russellian logicism – although the infinitary context would perhaps have been repugnant to Russell’s nominalistic mind. With respect to Fregean logicism there has been made, in the last decades, much progress concerning weakenings of the wicked Grundgesetz V, for instance to what is called Hume’s Principle. Another line of research is suggested in [1] where a system named pra¨ Kid is defined which retains the full Grundgesetz V, but weakens comprehension. The consistency of pra¨ Kid is still open but plausible.

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Notes 1.

2.

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4. 5. 6.

7. 8.

9.

10.

11. 12.

13.

That Frege’s logicism encompasses at least second-order arithmetic, and is bounded from above by, say, fourth-order arithmetic, can only be gleaned from the contents of [6]; there is no explicit demarcational statement made by Frege, neither for “arithmetic" nor for “logic" (of course not). On p. V we read: “The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify.” This statement implies a fortiori that logic is part of mathematics; we may call this Reverse Logicism. There is, up to now, no literature about reverse logicism. I have given several proofs of (or arguments for) weak[LT]. The first proof appeared in [1] and was made much clearer in [4]. These two proofs make use of a cumulative type structure, which is rather different from the type structure of the system Σℵ1 to be introduced below. The proof in the present paper was first outlined in [2], and detailed in [3]. Nevertheless, I think the representation in the present paper contains a deeper conceptual analysis which goes beyond that in [2] and [3]. Or rather a class of similar languages. Here equivalence may be isomorphism, or L-elementary equivalence, or something like that. It is an open and vexing problem to give a precise sense to the notion of an analytic statement (“Urteil"), in contradistinction to synthetic statements. I think that to be in V alid(L) for some rich model-theoretic logic L is a good candidate for an explication of analyticity. Recall that Frege in [5] and [6] wanted to show that arithmetical statements are analytic, contrary to Kant for whom they were synthetic. In a sense, Frege’s logicism with respect to arithmetic simply says that arithmetic theorems are analytic. But Frege gave no definition of analytic, except by equating analytic = purely logical. By the way, it is difficult to tell what Kant would have subsumed under the title arithmetic beyond sentences like 7 + 5 = 12. Observe that if this sentence is synthetic, then the sentence 12 = 7 + 5 should be analytic, according to Kant’s definition. It is more than enough to consider κ < ℵω . Beyond that, no essentially new results seem to be provable. However, to avoid oddnesses, κ is always assumed to be a regular cardinal. This typed predication −(− − −) may also be conceived of as a typed membership relation; from the standpoint of a many-sorted first-order logic, −(− − −) appears as a non-logical notion. But within a type-theoretic framework, −(− − −) is usually considered as a logical notion. As I said above, I decide such questions about the intrinsic logicality by fiat. The cut rule itself may be regarded as a non-logical rule in so far as it does not introduce logical signs. Therefore, strictly logical systems should not contain the cut rule; but they should perhaps be closed under the cut rule. Considering the vast variety of rules possible in the framework of sequent calculi (e.g. the topic of substructural rules) there is simply no criterion to tell logical and non-logical rules apart, if we shun semantic criteria. For κ = ℵ0 , this semantic incompleteness follows from G¨odel’s incompleteness theorem and the categoricity of the second-order Peano axioms. The other cases can be shown quickly as follows. Let κ > ℵ0 be a regular cardinal. We forget all κ-types that contain 0, i.e. we start our inductive definition of the κ-types with [ ]. Then there is exactly one standard model Mκ for this restricted type structure. We can formulate the continuum hypothesis CH adequately by using the restricted language (Hint: use the type [[ ][ ][ ] . . .]). Now suppose Σκ + (cut) were complete with respect to StdLκ ; since CH is either true of false in Mκ , then either the sequent CH =⇒ or the sequent =⇒ CH should be provable. However, this alternative can be refuted by adapting known settheoretic results. Note that for κ > ℵ0 G¨odel’s incompleteness theorem cannot be used to show that Σκ + (cut) is incomplete with respect to StdLκ since all these systems contain complete arithmetic, and are, of course, not recursively axiomatizable. For instance, the Frege-Russell natural number 2[[0]] is λX [0] .∃x, y(X(x) ∧ X(y) ∧ ¬x = y ∧ ∀z(X(z) → z = x ∨ z = y)). Clearly, the cut elimination property implies consistency. However the consistency of Σℵ0 + (cut) is provable in a fragment of primitive recursive arithmetic. By G¨odel’s second incompleteness theorem, the cut elimination property for Σℵ0 cannot be proven in even in full P M . This shows that the cut elimination property of a logical system may be vastly stronger than its consistency. It is well known that BD (= Borel Determinacy) is not provable in Zermelo’s set theory together with AC, hence not in P M + AC. From time to time I try to prove BD in Σℵ1 .

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14. It seems to be a desideratum in the history of mathematical logic to describe and assess the real mathematical content contained in the printed pages of the book Principia Mathematica. 15. Thus, e.g., ZF may not be mathematics since it contains too much platonic nonsense. Perhaps, by the same token, not even Z is (real) mathematics, and so on. In [4] one can find logical systems which embed stronger and stronger extensions of Z. 16. There seems to be a problem about the exact date of publication of the second edition of PM; my copy has the date 1927, for each of the three volumes. But elsewhere I found dates like 1923–27, and 1925–27. 17. One formulation of this principle is: “Whatever involves a l l of a collection must not be one of the collection.” [14] p. 37. The main source of ambiguity lies in the muddle-headed term involves. A thorough analysis of the vicious-circle principle has been given by G¨odel in his [7]. 18. See [10] p. 207: “Formally it [i.e. Ramsey’s version of P M ] is almost unaltered; but its meaning [my emphasis] has been considerably changed.” In view of this statement, the hard words in (1) above seem to be grossly exaggerated. 19. See [15] 5.502 and 6, where there is put no finite bound on the number of arguments.

References [1] J. W. Degen : Systeme der kumulativen Logik, Philosophia Verlag, Munich (1983) [2] J. W. Degen : Two formal vindications of logicism, in: Philosophy of Mathematics, Proceed. 15th Intern. Wittgenstein Symp. ed. J. Czermak, Wien (1993), 243–250 [3] J. W. Degen : Complete infinitary type logics, Studia Logica 63 (1999), 85–119 [4] J. W. Degen and J. Johannsen : Cumulative higher-order logic as a foundation for set theory, Math. Log. Quart. 46, 2 (2000), 147–170 [5] G. Frege : Die Grundlagen der Arithmetik, Breslau (1884) [6] G. Frege : Grundgesetze der Arithmetik, Jena (1893, 1903) [7] K. G¨odel : Russell’s Mathematical Logic, in: Philosophy of Mathematics, eds. P. Benacerraf and H. Putnam, Englewood Cliffs (1964) [8] Jean van Heijenoort : From Frege to Go¨ del. A Source Book in Mathematical Logic, 1879–1931, Harvard University Press (1967) [9] G. Nakhnikian (ed.) : Bertrand Russell’s Philosophy, Duckworth (1974) [10] F. P. Ramsey : Foundations, ed. Mellor, Routledge & Kegan Paul (1978) [11] B. Russell : The Principles of Mathematics. 2nd edition, Norton & Company (1938) [12] B. Russell : Mathematical logic as based on the theory of types, American Journal of Mathematica 30 (1908), 222–262, also in: [8] [13] K. Sch¨utte : Proof Theory, Springer (1977) [14] A. N. Whitehead and B. Russell : Principia Mathematica. 2nd edition, Cambridge (1927) [15] L. Wittgenstein : Tractatus logico-philosophicus, Hrsg. von B. McGuinness u. J. Schulte, Suhrkamp (1989)

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