Abstract
Leon Horsten has recently claimed that the class of mathematical truths coincides with the class of theorems of ZFC. I argue that the naturalistic character of Horsten’s proposal undermines his contention that this claim constitutes an analogue of a thesis that Daniel Isaacson has advanced for PA. I argue, moreover, that Horsten’s defence of his claim against an obvious objection makes use of a distinction which is not available to him given his naturalistic approach. I suggest a way out of the objection which is in line with the naturalistic spirit of Horsten’s proposal but which further weakens the analogy with Isaacson’s Thesis. I conclude by evaluating the prospects for providing an analogue of Isaacson’s Thesis for ZFC.
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Notes
Consider a series generated as follows: starting with the exponential base representation of a number n, the next term of the series is obtained by increasing each digit of the current base by 1 and subtracting 1 from the resulting number. Goodstein’s Theorem says that such a series tends to 0. See Goodstein (1944).
Goodstein’s proof makes use of induction up to ɛ0, which was shown by Gentzen (1936) to be adequate for proving the consistency of PA and is therefore, by Gödel’s Incompleteness Theorem, unavailable in PA. The unprovability of Goodstein’s Theorem in PA was then established by Kirby and Paris (1982) by using countable non-standard models of PA and the model-theoretic technique of indicators.
Horsten (2001, p. 175) notices that “if we want to be a little more precise, then we would have to say that [...] ZFC, when interpreted in the normal way, provides a good model of the set of mathematical truths.” The qualification is presumably needed to prevent the Conjecture from being trivially false, at least from the naturalistic point of view he endorses (see below), since for a naturalist mathematical truths are sentences of the informal language of mathematics. In the remainder of the paper I will follow Horsten in dropping the qualification.
Horsten considers this option. See Horsten (2001, p. 185).
Although this claim needs some further qualifications, which I will discuss in Sect. 5.
Friedman has conjectured that his work on Boolean Relation Theory will eventually lead mathematicians to accept large cardinal axioms by producing statements whose proof requires such axioms and that appear natural from the point of view of the working mathematician. See e.g. Friedman (2000). For a brief account of Boolean Relation Theory, see Friedman (2000, pp. 438–450).
See e.g. Horsten (2001, p. 189), where he writes that “mathematical truths have to be regarded as truths that are derivable from basic mathematical axioms.”
The qualification that a principle, beyond being accepted by the mathematical community, also needs to be mathematical is needed for reasons which will become clear in the next section.
Beyond Goodstein’s Theorem and GPA, examples include the Paris-Harrington (1977) variant of the finite Ramsey Theorem, Friedman’s finitization of Kruskal’s Theorem (Smorynski 1982) and various versions of the ω-rule. See Isaacson (1987, p. 217), Isaacson (1987, p. 219), and Isaacson (1992) respectively for arguments for the non-arithmeticity of these statements.
In particular, we need to add ‘ + ’ and ‘·’ to the primitives of the language and the recursion equations for addition and multiplication to the axioms of the theory. Gödel (1931) established that adjoining addition and multiplication to the base theory is sufficient to obtain a theory which is adequate to express all the primitive recursive and indeed general recursive functions.
See Isaacson (1992, p. 100).
Later in the paper, we will discuss cases in which the mathematical community accepts a statement as true for non-mathematical reasons.
For ease of exposition, I restrict attention to GZFC. Analogous considerations can be offered for (the provably equivalent) ConZFC.
Proof The axioms of ZFC are true. The rules of inference of ZFC are truth-preserving. Hence all the theorems of ZFC are true (Tarski 1935). Thus, \({\hbox{Tr}}({\hbox{ZFC}})\vdash \hbox{Prov}_{{\hbox{ZFC}}}(\varphi)\rightarrow {\hbox{T}}(\varphi),\) for all \(\varphi \in {\mathcal{L}}_{{\hbox{ZFC}}}\) (where ‘ProvS’ is the canonical predicate expressing provability in some system S). Hence, in particular, \({\hbox{Tr}}({\hbox{ZFC}})\vdash \hbox{Prov}_{{\hbox{ZFC}}}({\hbox{G}}_{\hbox{ZFC}}) \rightarrow {\hbox{T}}({\hbox{G}}_{\hbox{ZFC}}).\) By disquotation, \({\hbox{Tr}}({\hbox{ZFC}})\vdash \hbox{Prov}_{{\hbox{ZFC}}}({\hbox{G}}_{\hbox{ZFC}}) \rightarrow {\hbox{G}}_{\hbox{ZFC}}.\) Since we already knew by construction that \({\hbox{ZFC}} \vdash \neg \hbox{Prov}_{{\hbox{ZFC}}} ({\hbox{G}}_{\hbox{ZFC}}) \rightarrow {\hbox{G}}_{\hbox{ZFC}},\) we can then use proof by cases to show that \({\hbox{Tr}}({\hbox{ZFC}})\vdash (\hbox{Prov}_{{\hbox{ZFC}}}({\hbox{G}}_{\hbox{ZFC}})\lor \neg \hbox{Prov}_{{\hbox{ZFC}}}({\hbox{G}}_{\hbox{ZFC}})) \rightarrow {\hbox{G}}_{\hbox{ZFC}}.\) Since each instance of excluded middle is provable in Tr(ZFC), we thus have \({\hbox{Tr}}({\hbox{ZFC}})\vdash {\hbox{G}}_{\hbox{ZFC}}.\) ■
Proof One of the reflection principles added to the theory is \( \hbox{Prov}_{{\hbox{ZFC}}}({\hbox{G}}_{\hbox{ZFC}}) \rightarrow {\hbox{G}}_{\hbox{ZFC}}.\) Hence, trivially, \({\hbox{Rfn}}({\hbox{ZFC}}) \vdash \hbox{Prov}_{{\hbox{ZFC}}}({\hbox{G}}_{\hbox{ZFC}}) \rightarrow {\hbox{G}}_{\hbox{ZFC}}\) We can then reason as above. ■
See Horsten (2001, p. 177).
See Horsten (2001, pp. 177–178). Of course, given Horsten’s account of mathematical truth, it is nonetheless crucial for GZFC not to count as mathematical truth that the mathematical community does not accept it as a mathematical principle.
It might be objected that the proof of GZFC formalizable in Rfn(ZFC) only employs the concepts of provability in ZFC and consistency of ZFC, which are not philosophical. However, this worry can be overcome by thinking of the distinction as a distinction between mathematical and non-mathematical concepts. In what follows, I will stick to Horsten’s terminology and talk of philosophical versus mathematical concepts, since nothing relevant to the discussion hinges on this choice.
Compare Horsten (2001, pp. 181–182), where he writes that “for Isaacson the notion of arithmetical truth is in part an epistemological notion” and adds that “all that really matters for the [Conjecture] is that mathematical truth is, in part, an epistemological notion.”
In line with Horsten’s proposal concerning ZFC, I am assuming that S will be a consistent and recursively axiomatizable system, that it will contain elementary arithmetic, and that its provability predicate will satisfy the so called Hilbert-Bernays provability conditions. Taking for granted S’s consistency, the assumption that S will be recursively axiomatizable may be supported by the idea that the sentences accepted by mathematicians will be finitely derived from axioms and rules of inference and will therefore form a recursively enumerable set. By Craig’s lemma (Craig 1953), we then have that this set of sentences will be recursively axiomatizable. The assumption that S will contain elementary arithmetic is hardly disputable when one is concerned with the theory which the community of mathematicians accepts. As far as the third assumption is concerned, the idea that a provability predicate which does not satisfy the Hilbert-Bernays conditions does not even count as a provability predicate has been argued for by Kreisel and Levy (1968) and has been recently defended by Roeper (2003).
Thanks to an anonymous referee here.
For a convincing defence of this point, see Smith (2007, pp. 249–251). It is perhaps worth stressing that when one says informally that reflection principles express the idea that if φ is provable in S, then φ is true, one is using “true” purely disquotationally and for mere grammatical reasons. In spelling out the content of a reflection principle, we do not need to refer to φ, but only to state it, and are therefore making no essential use of the concept of truth. This is confirmed by the fact that the proof by way of reflection principles can be formalized without introducing a truth-predicate. For more on this, see Dummett (1994b).
To preserve the analogy with Horsten’s distinction between mathematical and philosophical concepts, I will talk of the community of philosophers as opposed to the community of mathematicians. The points I will make, however, would go through even if we were to talk of other communities instead, such as the community of logicians.
There might be yet other ways of explicating the distinction between philosophical and mathematical reasons. However, the differences that I highlight below between the Philosophical Community Thesis and the Philosophical Concepts Thesis are likely to remain whenever the distinction is explicated in naturalistically kosher terms and in such a way that GS is then accepted only for non-mathematical reasons.
GZFC can be proved in \({\hbox{ZFC}}+`\exists\) an inaccessible cardinal’. This is easily seen by noticing that an inaccessible cardinal is one that is closed under exponentiation and limits of smaller cardinals. It follows that if \(\kappa\) is inaccessible, then \(V_{\kappa}\) is a model of ZFC. Thus, if there exists an inaccessible cardinal ConZFC is a consequence, and so is the provably equivalent GZFC.
That is, given any two models of this theory one is isomorphic to an initial segment of the other. The result was first proved in Zermelo (1930).
McGee (1997) has pointed out that if we take the first-order quantifiers of ZFC2 as ranging over everything and add an axiom stating that the individuals form a set, we obtain a theory—call it ZFCU2—which is fully categorical as far as the pure sets are concerned. This, it might be argued, shows that the contrast between Isaacson’s Thesis and Horsten’s Conjecture on this point is not as stark as it appears at first sight. However, McGee’s result depends on the addition of extra assumptions which do need philosophical argument in order to be justified. Hence, the result does not per se show that the disanalogy between PA and ZFC on this point does not hold and at best points to a way to go about dealing with the said disanalogy.
It is known that \({\hbox{ZFC}}+`\exists\) a measurable cardinal’ proves the negation of the Axiom of Constructibility (Scott 1961) and, as already mentioned, that \({\hbox{ZFC}}+`\exists\) an inaccessible cardinal’ proves GZFC.
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Acknowledgements
Many thanks to Michael Potter, Peter Smith, and two anonymous referees for this journal. An earlier version of this material was presented at the workshop Arithmetical Truth—20 years on, held at Fitzwilliam College, Cambridge. Special thanks to Michael Potter for inviting me to the workshop and to the audience for many useful comments. Research for this paper was funded by St. John's College, the AHRC, and the Cambridge European Trust. I gratefully acknowledge the support of these institutions.
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Incurvati, L. Too Naturalist and Not Naturalist Enough: Reply to Horsten. Erkenn 69, 261–274 (2008). https://doi.org/10.1007/s10670-008-9114-1
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DOI: https://doi.org/10.1007/s10670-008-9114-1