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From: Torgny Tholerus <torgny.domain.name.hidden>

Date: Tue, 20 Nov 2007 19:01:38 +0100

Bruno Marchal skrev:

*>
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*> But infinite ordinals can be different, and still have the same
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*> cardinality. I have given examples: You can put an infinity of linear
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*> well founded order on the set N = {0, 1, 2, 3, ...}.
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*> The usual order give the ordinal omega = {0, 1, 2, 3, ...}. Now omega+1
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*> is the set of all ordinal strictly lesser than omega+1, with the
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*> convention above. This gives {0, 1, 2, 3, ... omega} = {0, 1, 2, 3, 4,
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*> ....{0, 1, 2, 3, 4, ....}}. As an order, and thus as an ordinal, it is
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*> different than omega or N. But as a cardinal omega and omega+1 are
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*> identical, that means (by definition of cardinal) there is a bijection
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*> between omega and omega+1. Indeed, between {0, 1, 2, 3, ... omega} and
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*> {0, 1, 2, 3, ...}, you can build the bijection:
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*>
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*> 0--------omega
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*> 1--------0
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*> 2--------1
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*> 3--------2
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*> ...
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*> n ------- n-1
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*> ...
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*>
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*> All right? "-----" represents a rope.
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*>
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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...

Date: Tue, 20 Nov 2007 19:01:38 +0100

Bruno Marchal skrev:

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 --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list.domain.name.hidden To unsubscribe from this group, send email to everything-list-unsubscribe.domain.name.hidden For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---Received on Tue Nov 20 2007 - 13:01:48 PST

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