jane - Reflections on Skolem's Relativity of...

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Reflections on Skolem's Relativity of Set-Theoretical Concepts! IGNACIO JANE" From 1922 onwards Skolem maintained that set-theoretical concepts are relative (in a sense of 'relative' that we must discern). In 1958 he viewed all mat.homatlcal notions as relative. The main instrument he used in his argument for set-theoretical relativity is the Uiwenheim-Skolem theorem, in the form that every consistent first-order countable set of sentences has a countable model. In 1938 he said of this theorem that 'its most important application is the critique of the set-theoretical concepts, and most espe- cially that of the higher infinite powers' (Skolem [1941], p. 460).1 My goal is to review Skolem's argument, chiefly as it appears in Skolem [1922], with the aim of gaining some insight into the ontology of set theory. Although I will often quote Skolem and try to be true to his words, what follows is not put forward as a faithful reconstruction of Skolem's actual view, but rather as an attempt to present it as a sensible one. Skolem is commonly portrayed as arguing that certain otherwise well un- derstood concepts are suspect simply because they cannot be characterized in a first-order language; in particular that, since all first-order formaliza- tions of set theory (if consistent) have countable models, the concept of uncountability is flawed. I hope to show that Skolem's position is more solid than that. I see Skolem as arguing that all the evidence that has been given for the existence of uncountable sets is inconclusive, and the reason why he insists on considering countable models is that axiomatization was put forward at the time as the only way to secure set theory, and what sets are and which sets exist was claimed to be determined by the axioms and their models (much as what Euclidean geometry is about was claimed to be determined by Hilbert's axioms and their models). this situation, bringing countable models into play was perfectly in order, all the more so as no other models could be supplied without set-theoretical means. Today t Part.ially supported by Spanish DGICYT grants PS04-0244 and PB97-0948. Doparuuuoru de I. .b/;ica, lllstorla I Filosofla de la Ciencia, Universitat de Barcelona, Haldiri Hcrxa« S/II, E. .08028 Harcelona, Spain. jane@mat.ub.es. I 'SOli upplication la plus importante est, en effcl, la critique des concepts de la theorie des ensembles, ct plus spocialemcnt celie des puissances infinies superieures.' PlllLOSOPllIA MATIIEMATICA (3) Vol. 9 (2001), PI'. 129-153.
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JANE we may no longer uphold this claim, but if we do believe that there are uncountable sets, we should be willing to comply with Skolern's require- ment that their existence be substantiated by some means other than mere formal postulation. Replies to Skolem usually take for granted that there are uncountable sets (and, of course, if there are, then our inability to characterize them with certain limited means is only a sign of the inadequacy of these means). They remind us that set theory is really about sets and then proceed to argue
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This note was uploaded on 08/01/2008 for the course PHIL 140A taught by Professor Fitelson during the Spring '07 term at University of California, Berkeley.

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jane - Reflections on Skolem's Relativity of...

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