Cantor's Paradise on Skolem's Earth
Mangesh Padwardhan (National Insurance Academy Pune)
As a matter of conceptual coherence, the study of set theory and its models cannot get off the ground unless we have a stock of sets that are already available to us; as set or class models of set theory are made up of, well, sets themselves or their collections. Therefore, it seems that not only universists but even radical multiversists like Hamkins have to reckon with the need for such stock of sets. In recent years, the iterative set theoretic hierarchy “V” has been widely accepted as the stock of all sets that are there, in some sense. However, it raises several issues. One, as Quine remarked, this conception seems to carry staggering ontologi- cal presuppositions. Two, there is the issue of width and height potentialism. Proponents of width potentialism such as Feferman maintain that the concept of powerset is vague, even at the “lowly” level of the continuum. The issue whether giving a second order characterization of set theory and thereby V involves an illicit and viciously circular appeal to the powerset concept itself continues to be debated. Even leaving that aside, the problem of height po- tentialism remains. There is no principled way to decide where one should stop iterating the powerset operation and take the collection of sets constructed till then as V (provided it is a strongly inaccessible rank). As Maddy remarks, it is difficult to say why the “powersets” atop V is not just another stage we forgot to include. In fact, Zermelo visualised the set theoretic universe as an unlimited progression of models of set theory with no true end, but only rela- tive stopping points. The reformulation by Shepherdson (endorsed by Isaacson) of Zermelo’s argument as proof in first order ZF regarding a class of full inner models of NBG is philosoph- ically unsatisfactory. It can also be argued that the existence of large large cardinals does not follow from the iterative picture. Other sophisticated philosophical considerations such as uniformity, generalization and inexhaustibility have been brought in to justify these. I pro- pose that we turn Skolem’s criticisms of set theory and in particular “Skolem’s paradox” on their head to get an initial stock of sets and get the model theory of set theory going. His “numerical model” version - if an axiom system A (such as ZFC) is consistent, it has a model in natural numbers – fits the bill. This approach should be acceptable to those troubled by thestaggering ontology implicit in the iterative universe picture as well as by potentialism issues. Moreover, it should be unproblematic to Maddy’s second philosopher. Skolem’s contention that in light of his analysis, the theorems of set theory can be made to hold in a mere verbal sense actually becomes a virtue. I argue that this formulation allows us to enjoy the beauty of Cantor’s paradise while remaining grounded on Skolem’s earth.