℘(ℵ₀) ≠ ℘(ℵ₁) is not provable in ZFC (this unprovability is an instance of Easton's theorem). I don't know why my mind decided to get hung up on this today, but I'm tired and this brain bee is buzzing like mad in my head.
My confusion goes:
- Take the well-founded set of all subsets of ω. This set is not itself a subset of ω, so it has some elements that are not elements of ω. (So we have Cantor's theorem streamlined by the foundation axiom.)
- All elements of subsets of ω are finite ordinals. There are subsets with countably many such elements (e.g. the subset consisting in all the even ordinals, or all the finite ordinals besides 0, or all besides 0 and 1, etc.), but no specific element is itself countably infinite (much less uncountably so).
- But ω1 has subsets with elements that are individually countably infinite.
- So ω1 has subsets that ω doesn't have.
- So shouldn't ℘(ω) be non-identical to ℘(ω₁)?
All I can think of, as to where I'm off-key, is that non-identity is not the same as inequality, here; of course the one powerset is not strictly identical to the other, but they could still be equal in size and so ℘(ℵ₀) = ℘(ℵ₁) would be thereby possible. However, my general sense of the "greater than" relation, here, is mediated by my sense of the foundation axiom; so having foundation over ℘(ω₁) involve different subsets than over ℘(ω) motivates me to think that the one has to be greater than the other.
But then alternatively:
- ZFC ⊨ (ZFC ⊭ (℘(ℵ₀) ≠ ℘(ℵ₁)))
- ZFC ⊨ (ZFC ⊨ (℘(ℵ₀) ≠ ℘(ℵ₁)))
But then I would have shown that a contradiction can be derived from ZFC, which I would be surprised to have accomplished one random morning while groggy and irritated. On the other hand, in a response to a MathOverflow question about ZFC's consistency, Hamkins commented that:
What I heard (and this is all second hand) was that he [Silver] thought (and I agree with the concern) that we might be making a fundamental logical mistake in the too-often conflation of internal object theory notions with metatheoretic notions. For example, Con(ZFC) uses the object theory notion, but we are thinking about the metatheory. To my way of thinking, this is related to the reluctance of many large cardinal set theorists to take illfounded models seriously, since insisting on well-founded models is exactly to identify the object theory notions with the meta-theoretic notions. [emphasis added]
Not that the comment is directly pertinent to my confusion, but it seems to indicate that if ZFC were to be proven inconsistent, it would be modulo the foundation axiom somehow. So maybe, grogginess and irritation aside, I would have hit upon such a proof? (I really doubt it, though.)
