Axiomatization and Incompleteness in Arithmetic and Set Theory

Wesley Duncan Wrigley

I argue that are (at least) two distinct kinds of mathematical incompleteness. Part A of the thesis discusses Gödelian incompleteness, while Part B is concerned with set-theoretic incompleteness. Both parts are concerned with the philosophical justification of reflection principles and other axiomatic devices which can be used to reduce incompleteness, and in particular with the justification of such devices from the philosophical standpoint of Kurt Gödel.

In Part A I consider Gödel's disjunctive argument. In chapter 1, I argue that the non-mechanical mind considered by Gödel is best modelled by a theory constructed using the transfinite iterated application of a soundness reflection principle to PA. I argue that Feferman's completeness theorem shows this account of the mind to be incompatible with some elementary assumptions in the epistemology of arithmetic. In chapter 2, these considerations are developed into a positive argument for the existence of absolutely undecidable arithmetical propositions. The consequences for the indefinite extensibility of the concept natural number are then discussed. I argue that properly understood, Feferman's theorem refutes Dummett's position in the debate.

I begin part Part B in chapter 3, by reconstructing a version of Gödel's platonism, called conceptual platonism. I then examine how such a position relates to various means of reducing set-theoretic incompleteness. In chapter 4 I argue that there is some prospect for this position of effecting a limited reduction in incompleteness by means of reflection principles justified by mathematical intuition. However, such priniciples are incompatible with Gödel's commitment to platonism about properties of properties of sets. In chapter 5 I argue that conceptual platonism does not lend support to the view that a substantial reduction in incompleteness can be effected by large cardinal axioms justified using extrinsic methods analogous to the principles of theory choice in natural science. This undercuts the traditional justification for many large cardinal axioms, so I end with a sketch of how conceptual platonism could be modified to rehabilitate the large cardinals program.