That's is literally what I said. My post:
I'm saying that even as metaphysical dialetheist I do not believe a "barber who shaves all and only those who do not shave themselves" cant exist.
— Me — MindForged
Are you serious? Your OP:
The town barber, who is a man, shaves exactly every man in the town who does not shave himself.
— Jeremiah — MindForged
As I say, that's a somewhat naive view. The specification scheme allows one to avoid the paradox, but it doesn't necessarily solve the paradox. The whole point of regimenting set theory this way was to make make math consistent (or at least not provably inconsistent). But it comes with well known issues, like a number of unsolved questions that have known answers in other systems (e.g. Continuum Hypothesis). — MindForged
MindForged, you are completely misunderstanding the difference between a veridical paradox and a plain old proof by contraction. Moreover, Russell's paradox has absolutely nothing to do with Gödelean incompleteness. Simply nothing. — fishfry
The town barber, who is a man, shaves exactly every man in the town who does not shave himself.
Does the barber shave himself? — Jeremiah
Claim: We cannot form sets out of arbitrary predicates.
Proof:
Assume the negation of our claim: That is, assume that we can always form a set out of a predicate.
Consider the predicate P(x) = "x ∉ x".
Now we let R be the set R = {x : P(x)}. We see (following Russell) that we must have both R ∈ R and R ∉ R. That's a contradiction. — fishfry
Let R be the set of all sets that are not members of themselves
Is R a member of itself? If so, then it must meet the condition of not being a member of itself, which would mean it is not. If it is not, then it must meet the condition of not being a member of itself, which would mean it is a member of itself. — Jeremiah
You do realize argument from authority is only wrong if the authority is wrong. I think over 100 years of history is a very strong authority. — Jeremiah
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