On Forms of Justification in Set Theory

Abstract

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how ‘intuitively plausible’ an axiom is, whereas extrinsic justification supports an axiom by identifying certain ‘desirable’ consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation.

What Model Companionship Can Say About the Continuum Problem

“…Therefore, these results single out (a definable version of) 2 ℵ 0 = ℵ 2 as the unique solution of the continuum problem which can fall in at least one AMC of set theory enriched with large cardinal axioms (for example the AMC with respect to signature * of item (iv) above). 9 The present model-theoretic approach to set theoretic validities complements well standard strategies for producing axiomatic extensions of ZFC. Actually we see our approach as an implementation of G ödel's program.…”

“…Intrinsic and extrinsic reasons have been extensively studied in the philosophy of set theory [29, 30] and they have been widely applied, in recent debates, for the justifications of competing programs [2, 23]. These two forms of justification have also been criticized for their opacity in offering clear criteria of application and for the lack of demarcation between intrinsic and extrinsic reasons [9]. Moreover, as it happened in the debate on the analytic-synthetic distinction, there is also no shortage of contributions that reject the problem of justification at its very base.…”

“…8 For example T could have CH or 2 ℵ 0 > ℵ 2 among its axioms provided it extends ZFC+there are class many supercompact cardinals. 9 Furthermore the same argument we used for CH and applies mutatis-mutandis to any set theoretic statement which can be formalized by a Π 2 -sentence in the signature * of item (iv) and such that is forcible over any model of set theory enriched by large cardinal axioms.…”