On Forms of Justification in Set Theory

Abstract

In the contemporary philosophy of set theory, discussion of new axiomsthat purport to resolve independence necessitates an explanation of howthey come to bejustified. Ordinarily, justification is divided into two broadkinds:intrinsicjustification relates to how ‘intuitively plausible’ an axiomis, whereasextrinsicjustification supports an axiom by identifying certain‘desirable’ consequences. This paper puts pressure on how this distinctionis formulated and construed. In particular, we argue that the distinction asoften presented is neitherwell-demarcatednor sufficientlyprecise. Instead, wesuggest that the process of justification in set theory should not be thoughtof as neatly divisible in this way, but should rather be understood as a con-ceptually indivisible notion linked to the goal ofexplanation.

What model companionship can say about the Continuum problem

“…Let τ be a signature. 7 Moreover, none of the signatures for group theory that we presented in Section 2 have existentially closed models which give rise to an elementary classe It is useful to recall that model complete theories are axiomatized by their Π 2 -consequence (see [10,Prop. 3.5.10]), and that the model companion of a τ -theory T (if it exists) is unique (by [10,Prop.…”

“…Intrinsic and extrinsic reasons have been extensively studied in the philosophy of set theory [23,24] and they have been widely applied, in recent debates, for the justifications of competing programs [2,20]. 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 [7]. Moreover, as it happened in the debate on the analyticsynthetic distinction, there is also no shortage of contributions that reject the problem of justification at its very base.…”