Conclusion

Abstract

The usual platonist argument against the intuitionistic, and for the classical, iterative conception of the set-theoretic universe¹ is the following. Just as with potential infinity, which the intuitionist accepts, we can stretch our concept of ‘possibility’ so that, for example, the power set operation — the ‘construction’ of all the subsets of a set — is well-founded at any stage in the hierarchy. That is, our concept of ‘construction’ (or ‘constructible’) can be extended, so that we can understand the power set axiom applied to an infinite set by analogy with the same axiom applied to a finite set. Our understanding of the actually, or of the uncountably, infinite, and of arbitrary infinite sets, is therefore possible by thinking of them as analogous to finite sets. For they are produced by taking axioms which are clearly correct for finite sets, and extending their application to infinite sets.