Arithmetical Knowledge and Arithmetical Definability: Four Studies

The subject of this dissertation is arithmetical knowledge and arithmetical definability. The first two chapters contain respectively a critique of a logicist account of a preferred means by which we may legitimately infer to arithmetical truths and a tentative defense of an empiricist account. According to the logicist account, one may infer from quasi-logical truths to patently arithmetical truths because the arithmetical truths are representable in the logical truths. It is argued in the first chapter that this account is subject to various problems: for instance, the most straightforward versions seem vulnerable to various counterexamples. The basic idea of the alternative empiricist account considered in chapter two is that complicated arithmetical truths like mathematical induction may be inferred by way of confirmation from less complicated quantifier-free arithmetical truths. The notion of confirmation here is understood probabilistically, and responses are given in this chapter to several seeming problems with this importation of probability into arithmetic.

The final two chapters are concerned with arithmetical definability in two different settings. In the third chapter, the interpretability strength of the arithmetical and hyperarithmetical subsystems of second-order Peano arithmetic is compared to the interpretability strength of analogous systems centered around two principles called Hume's Principle and Basic Law V, which respectively axiomatize a standard notion of cardinality and an alternative conception of set. One of the major results of this chapter is that the hyperarithmetic subsystem of Hume's Principle does not interpret the hyperarithmetic subsystem of second-order Peano arithmetic. The fourth chapter is concerned with arithmetical definability in the setting of descriptive set theory, where the relevant benchmark is between notions which may be defined without quantification over elements of certain topological spaces (Borel notions) and notions whose definitions do require such quantification (analytic, coanalytic, projective notions). In this fourth chapter the Denjoy integral is studied from the vantage point of descriptive set theory, and it is shown that the graph of the indefinite integral is not Borel but rather is properly coanalytic. This contrasts to the Lebesgue integral, which is Borel under this measure of complexity.