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Full Idea
The set-theory account of infinity doesn't just say that we can keep on counting, but that the natural numbers are an actual infinite set. This is necessary to make sense of the powerset of ω, as the set of all its subsets, and thus even bigger.
Gist of Idea
Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 5)
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.65
A Reaction
I don't personally find this to be sufficient reason to commit myself to the existence of actual infinities. In fact I have growing doubts about the whole role of set theory in philosophy of mathematics. Shows how much I know.
| 10183 | An infinite set maps into its own proper subset [Dedekind, by Reck/Price] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 14427 | We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell] |
| 9944 | We understand some statements about all sets [Putnam] |
| 9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |