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 --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---