@JadeVanadium: I always find it amusing that ZF, despite being immeasurably stronger than ACA0, has trouble with infinite sets without AC, whereas ACA0 has no trouble just because it already stipulates that every set is a subset of ℕ. Just for reference, here is the proof. Take any S⊆ℕ such that ∀k∈ℕ ( S does not biject with ℕ[<k] ). Let F = { ⟨k,m⟩ : k∈ℕ ∧ m∈S ∧ ℕ[<k] bijects with S[<m] }. Note that to get F within ACA0, we need to code ( ℕ[<k] bijects with S[<m] ). Then F bijects ℕ with S.

This is easy by induction. Clearly 0∈dom(F) since S ≠ ∅ so ⟨0,min(S)⟩∈F. And for any k∈dom(F), we have some m∈S such that ℕ[<k] bijects with S[<m], so letting n be min of S[≥m], we get ℕ[<k+1] bijects with S[<n] and hence k+1∈dom(F).

To be clear, ( ℕ[<k] bijects with S[<m] ) can be coded as ( there is a finite sequence x of length k such that i is an item in x iff i∈S[<m] ).

@user21820 It's definitely very silly. To be honest though, I think it's equally silly that ZFC still has so much trouble assigning cardinalities to uncountable sets, in the sense that CH and GCH are undecidable.

Hahaha.. that's if you believe in those monsters. =P

Aleph[1] stole my wife

LOL.

That statement has an implicit existential of "your wife". =P

I regret to confess that it is an implicit universal

Hahaha.. actually I'm not sure what native English speakers would interpret it as.

I am a native speaker, but I am still not sure.

Definitely implicit existential

I'm talking about a different ambiguity now. Does "X stole my Y" mean "There is some Z that I call my Y and X stole Z.", or does it mean "There is some Z that I used to call my Y and X stole Z."?

Like, did Aleph[1] steal your wife and hence make her no longer yours? XD

By the way, I would personally say that CH has multiple variants, depending on which notion of countable you use:

I am certain that the 1st and 3rd are distinct notions, and I believe the 1st and 2nd are equivalent for non-empty S. Either way, we have at least 2 choices for each side of CH, so we have 4 different variants...

As I mentioned to you before:

In particular, (ℕ→ℕ) is also weak-enumerable...

I have to go for a while now. Back later!

If you think about all 4 variants of CH, let me know what you think! =D

@JadeVanadium: I'm back!

@user21820 I think when someone says "stole my Y", the implication is that the speaker still consider Y to be 'theirs'. In the case of property, a thing being stolen doesn't legally change the fact that it's your property. Obviously people are not property though (ideally...), so something like "stole my wife" doesn't actually make sense when taken literally. The implication is that the speaker still considers her to be 'theirs' romantically, despite that the marriage is evidently dysfunctional.

XD

@JadeVanadium: I was away again, and am back again.

I didn't know the wife-stealing linguistics question was more interesting than the CH question. =P

Sorry I'm already focused on another math thing hahah

I'm about to post a question about whether the theory of (Ord, <) is decidable

Hahaha..

I think it is but I couldn't figure out a simple proof

Post the link here when you're done!

@JadeVanadium: I'm surprised. This is your first post on MO!

Hahah yeah. I guess that website is just weirdly intimidating

@JadeVanadium Ironically enough, it's also the website that is most likely to outlive the rest of SE-related sites if the SE management botches something even bigger, because MO moderators made some kind of agreement that they can at any time break off from SE and retain control over MO.

"if", more like "when"

I don't know the details, and it can't be easy to DIY, but the option is a little bit comforting.

@JadeVanadium HAHA. Please don't ask for it. XD

@JadeVanadium Looks like Emil answered it already.

XD

It didn't last as long as I thought. =P

Indeed. I didn't think it would take long, honestly

Although I'm not sure why they answered in a comment instead of posting an actual answer

I'm still reading through the article they referenced