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Re: Possible major blunder in Rado's version of Canonical Ramsey Theorem that goes far beyond omitting proof steps
Posted:
Nov 10, 2013 11:30 AM
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In message <BWpz$uCS15fSFw6Q@212648.invalid>, David Hartley
<me9@privacy.net> writes
>As far as I know (which isn't far) Ramsey's theorem only applies to
>countable sets so Rado's proof will only work with countable A.
That was a bit silly of me. For any A, if Ramsey's theorem applies to an
infinite subset B of A then it applies to A. So, if you're worrying
about choice, it is certainly true for at least any set with a
denumerable subset.
The statement and proof of Ramsey's theorem in the Erdos-Rado paper
apply only to sets of natural numbers. They can easily be extended to
any countably infinite set but not obviously to an infinite set with no
denumerable subset. So if you want the Canonical theorem to apply to
all infinite sets, neither proof works without at least a weak form of
choice.
I suspect that Rado's proof can be modified to show that if Ramsey's
theorem extended to all infinite sets holds in ZF, then so does the
extended canonical theorem, but I haven't checked it through carefully.
--
David Hartley
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