Bruno Marchal skrev:
>
> But infinite ordinals can be different, and still have the same
> cardinality. I have given examples: You can put an infinity of linear
> well founded order on the set N = {0, 1, 2, 3, ...}.
> The usual order give the ordinal omega = {0, 1, 2, 3, ...}. Now omega+1
> is the set of all ordinal strictly lesser than omega+1, with the
> convention above. This gives {0, 1, 2, 3, ... omega} = {0, 1, 2, 3, 4,
> ....{0, 1, 2, 3, 4, ....}}. As an order, and thus as an ordinal, it is
> different than omega or N. But as a cardinal omega and omega+1 are
> identical, that means (by definition of cardinal) there is a bijection
> between omega and omega+1. Indeed, between {0, 1, 2, 3, ... omega} and
> {0, 1, 2, 3, ...}, you can build the bijection:
>
> 0--------omega
> 1--------0
> 2--------1
> 3--------2
> ...
> n ------- n-1
> ...
>
> All right? "-----" represents a rope.
>
An ultrafinitist comment:
In the last line of this sequence you will have:
? --------- omega-1
But what will the "?" be? It can not be omega, because omega is not
included in N...
--
Torgny
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