Top News, Articles, and Interviews in Philosophy

Luca Incurvati’s Conceptions of Set, 7

Philosophy News image
In a couple of very well known papers, George Boolos argued that “the axioms of replacement do not follow from the iterative conception”. Was he right? Or can Replacement be justified on (some core version of) the iterative conception? This is the topic of the particularly interesting §3.6 (pp. 90–100) of Conceptions of Set, ‘The Status of Replacement’. Luca discusses three lines of argument to be found in the literature for the thought that the iterative conception does warrant Replacement. I’ll comment on two in this post. The first he calls Gödel’s Argument. Two quotes from Gödel: (i) “From the very idea of the iterative concept of set it follows that if an ordinal number α has been obtained, the operation of power set P iterated α times leads to a set Pα(∅)” And then (ii) “the next step will be to require that any operation producing sets out of sets can be iterated up to any ordinal number.” In response, Luca makes the following central points: Tait and Koellner have argued that elaborating Gödel’s claim (i) requires appeal to Choice. But not so. For we can work with the Scott-Tarski definition of an ordinal, and then, without needing an assumption of Choice, Gödel’s thought will at least warrant adding to Z+ the Axiom of Ordinals — the axiom that there is a level Vα for every ordinal α. This theory with the Axiom of Ordinals is rich, much more powerful than Z+, and in fact buys us the nice results that Boolos claimed for Replacement. However, the Axiom of Ordinals is weaker than full Replacement. A version of Gödel’s second claim (ii) is needed to get us from the iterative conception to full Replacement, and it isn’t clear why (ii) should be thought of as part of the iterative conception. On (1), accepting Gödel’s (i), Luca’s discussion seems spot on. On (2) quite a few readers (those familiar with ZFC but who haven’t read Potter’s book) might well have welcomed rather more at this point on the Axiom of Ordinals, on its. . .

Continue reading . . .

News source: Logic Matters

blog comments powered by Disqus