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# ‘Second-Order Languages and Mathematical Practice’, Journal of Symbolic Logic, 50, pp. 714-42

DOI link for ‘Second-Order Languages and Mathematical Practice’, Journal of Symbolic Logic, 50, pp. 714-42

‘Second-Order Languages and Mathematical Practice’, Journal of Symbolic Logic, 50, pp. 714-42 book

# ‘Second-Order Languages and Mathematical Practice’, Journal of Symbolic Logic, 50, pp. 714-42

DOI link for ‘Second-Order Languages and Mathematical Practice’, Journal of Symbolic Logic, 50, pp. 714-42

‘Second-Order Languages and Mathematical Practice’, Journal of Symbolic Logic, 50, pp. 714-42 book

## ABSTRACT

There are well-known theorems in mathematical logic that indicate rather profound differences between the logic of first-order languages and the logic of second-order languages. In the first-order case, for example, there is GodePs completeness theorem: every consistent set of sentences (vis-a-vis a standard axiomatization) has a model. As a corollary, first-order logic is compact: if a set of formulas is not satisfiable, then it has a finite subset which also is not satisfiable. The downward Lowenheim-Skolem theorem is that every set of satisfiable first-order sentences has a model whose cardinality is at most countable (or the cardinality of the set of sentences, whichever is greater), and the upward Lowenheim-Skolem theorem is that if a set of first-order sentences has, for each natural number n, a model whose cardinality is at least n, then it has, for each infinite cardinal k (greater than or equal to the cardinality of the set of sentences), a model of cardinality k . It follows, of course, that no set of first-order sentences that has an infinite model can be categorical. Second-order logic, on the other hand, is inherently incomplete in the sense that no recursive, sound axiomatization of it is complete. It is not compact, and there are many well-known categorical sets of second-order sentences (with infinite models). Thus, there are no straightforward analogues to the Lowenheim-Skolem theorems for second-order languages and logic.1 There has been some controversy in recent years as to whether “second-order

logic” should be considered a part of logic,2 but this boundary issue does not concern me directly, at least not here. The present approach is to assess the adequacy of first-order languages in formalizing actual mathematical practice. This problem is one that occupied mathematicians and logicians earlier this century (see Moore [1980]), but seems to have received less attention recently. My main conclusion, in agreement with Bernays, Hilbert, and Zermelo (and in disagreement with Godel and Skolem), is that no first-order language is sufficient for axiomatizing such branches

as arithmetic, real and complex analysis, and set theory-branches that (each) deal with a particular (infinite) structure or, in other words, branches whose languages have “intended interpretations”. The argument presented rules out any language whose logic is either complete or compact. I go on to suggest that nothing short of a language with second-order variables will do. The considerations brought against first-order languages are semantical. That is

to say, I argue that the semantics of first-order languages is not adequate for the preformal semantics of mathematical practice. A few brief comments elaborating this perspective are in order. First, the standpoint of this article might best be characterized as a “neutral

realism”. The “realism” indicates that mathematical discourse is taken at face value. Contra formalism, (most) mathematical assertions are regarded as meaningful assertions about mathematical entities. Mathematical truth is determined by the subject matter of mathematics and, thus, “truth” is synonymous with neither (real/ideal) “knowledge” nor “provability”. Contra intuitionism and logicism, no attempt is made to criticize the bulk of mathematical practice. Rather, actual mathematical practice is taken to be the data for the considerations of this paper. The “neutral” in “neutral realism” indicates that at present, I have no view on the makeup or the ontological status of the subject matter of mathematics. I only hold that there is such a subject matter. In fact, I have no a priori objection to any interpretation of mathematics as long as the integrity of the bulk of mathematical discourse is preserved.3 It might be noted that much (but not all) of the literature on both sides of the first-

order/second-order issue takes or presupposes a viewpoint of realism. For example, Godel, an avowed platonist, was one of the strongest (and probably the most influential) proponents of first-order languages. Moreover, the recent arguments (see footnote 2) against higher-order languages are directed at the semantics of such languages and, consequently, seem to presuppose a realism of sorts. Indeed, if one is concerned only with codifying mathematical proof (for example, if one is a formalist of the Hilbert or Curry school), then there is little to object to concerning any effectively specified language and deductive system. In short, if one is not concerned with the interpretation of a language of mathematics as such, then, a fortiori, one will not worry about such things as excess ontological commitment and inconvenient semantic properties. Higher-order languages are considered here with standard semantics in which, for

a given interpretation, the second-order predicate variables range over all of the subsets of the domain, the second-order function variables range over all of the functions from the domain to the domain, etc. There is, of course, an alternate semantics for second-order languages, developed originally in Henkin [1950], in which, for a given interpretation, the predicate variables range over a fixed subset of the power-set of the domain, etc. For this alternate semantics (and the usual deductive system), second-order logic is sound, complete, and compact. Although it will not always be demonstrated directly, it is easily seen that most of the present

considerations against first-order languages apply to second-order languages with Henkin semantics. Finally, the completeness theorem indicates that for first-order languages, the

proof theory corresponds in a direct way with the semantics, or model theory. Consistency and satisfiability are coextensive, as are deductive consequence and semantic consequence. This, of course, is not the case with second-order languages. Thus, present considerations concerning the semantics of mathematics do not shed much light on the question of which deductive systems are appropriate for codifying mathematical proof. To put it differently, my thesis is that reference to the predicates or subsets of given domains is necessary to capture the semantics of mathematical practice, but I have little to say concerning the particular axioms or assumptions about such subsets necessary to codify normal proof techniques.4 As noted, first-order languages do not allow categorical characterizations of

infinite structures. I take this as their main shortcoming. §1 deals with the importance of categoricity in understanding and communicating mathematics. This involves the relevance of the Lowenheim-Skolem theorems to the present issue and the epistemic presuppositions of second-order languages. §2 concerns further inadequacies of first-order versions of arithmetic, analysis, and set theory, concluding that such theories do not capture important, perhaps crucial, aspects of those fields. §3 is a discussion of the adequacy of several alternate languages. The first subsection concerns languages “intermediate” between first-order and secondorder; the second subsection concerns the language of first-order set theory. The final §4 is a brief comparison of the semantics and proof-theoretic strength of standard first-order logic with that of standard second-order logic. One of the purposes of logic is to codify correct inference. Thus, if my major

conclusions are correct, the underlying logic of many branches of mathematics is (at least) second-order: one cannot codify the correct inferences of a second-order language with a first-order logic. It follows that the inconvenient technical properties and presuppositions of second-order logic must be accepted. The correct conclusion, I believe, is that there is no sharp distinction between logic and mathematics. The study of correct inference, like almost any other science, involves some mathematics and some mathematical presuppositions.