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 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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