non-recursive countable ordinal
in [Math]
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From: master1729 on 17 Jun 2010 13:13 non-recursive countable ordinals ... do they really exist ?? (i already objected to ordinals before , but those objections were mainly towards uncountable ordinals) perhaps the final countable debate ....
From: Rupert on 17 Jun 2010 21:17 On Jun 18, 7:13 am, master1729 <tommy1...(a)gmail.com> wrote: > non-recursive countable ordinals ... > > do they really exist ?? > > (i already objected to ordinals before , but those objections were mainly towards uncountable ordinals) > > perhaps the final countable debate .... Well, there is no doubt that they can be proved to exist in ZF.
From: Bill Taylor on 18 Jun 2010 01:29 > > non-recursive countable ordinals ... > > do they really exist ?? > Well, there is no doubt that they can be proved to exist in ZF. Yes, I wonder though if anything can be said about any individual one, that is in any way constructive/definitional? Of course, one may refer to "the least nonrecursive ordinal", but that hardly resonates with the last 2/words in the above. Is there any other sort of referable one? I can't help feeling that there must be. I have also heard that UNcountable ordinals can be used in some way, to define further COUNTable ordinals - even recursive ones, perhaps? I'm not sure. Can anyone help elucidate how this is done, with a paragrahic outline? -- Well-ordered William * And God said * Let there be numbers * And there WERE numbers. * Odd and even created he them, * He said to them, be fruitful and multiply, * And he commanded them to keep the laws of induction
From: master1729 on 21 Jun 2010 10:27 > On Jun 18, 7:13 am, master1729 <tommy1...(a)gmail.com> > wrote: > > non-recursive countable ordinals ... > > > > do they really exist ?? > > > > (i already objected to ordinals before , but those > objections were mainly towards uncountable ordinals) > > > > perhaps the final countable debate .... > > Well, there is no doubt that they can be proved to > exist in ZF. well , an ordinal is countable if its cardinality is countable. thus if and only if there is a bijection between an infinite ordinal and the integers , that infinite ordinal is countable. but if that countable ordinal is not recursive , how can there be a bijection to the integers ? a bijection to the integers needs to exist , otherwise we can use cantors diagonal and show uncountability. thus , no non-recursive countable ordinal exists ? tommy1729
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