Talk:Jónsson cardinal
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Wrong definition?
[edit]The definition given appears to be wrong; it is equivalent to weak compactness. Ben Standeven 21:37, 27 June 2006 (UTC)
- I've looked it up in Kanamori, but I may be misremembering it. Ben Standeven 22:29, 16 August 2006 (UTC)
- I do not know about the old definition, but your new one cannot be right. If the range of the function is allowed to be κ, then the function can be chosen to be one-to-one and thus not homogeneous on any subset larger than cardinality 1 (certainly not κ). So I will revert you. JRSpriggs 02:33, 17 August 2006 (UTC)
- Yeah; I basically had it right, but the partition symbol he was using wasn't the standard one. Ben Standeven 22:29, 19 August 2006 (UTC)
- I do not know about the old definition, but your new one cannot be right. If the range of the function is allowed to be κ, then the function can be chosen to be one-to-one and thus not homogeneous on any subset larger than cardinality 1 (certainly not κ). So I will revert you. JRSpriggs 02:33, 17 August 2006 (UTC)
Did you mean to say "An uncountable cardinal number κ is said to be Jónsson if for every function f: [κ] < ω → κ there is a set H of order type κ such that for each n, f restricted to n-element subsets of H omits at least one value in κ." with subsets of H? Or is H now superfluous? JRSpriggs 07:35, 20 August 2006 (UTC)
- Oh, yeah; f omits values on H, not necessarily on kappa. I'll fix that. Ben Standeven 02:11, 24 August 2006 (UTC)
The definition given is not that in Kanamori. There, the range of f on [H] < ω omits at least one value in κ. That seems to be stronger than the definition given here, namely that the range of f on [H] n omits at least one value in κ, for each n. Perhaps they're equivalent. But I couldn't find a statement or proof of this in Kanamori or Jech. — Preceding unsigned comment added by 2001:9E8:9B33:E00:B9E2:DFCC:17B0:3193 (talk) 09:20, 11 September 2023 (UTC)