# RE: Bijections (was OM = SIGMA1)

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> Date: Tue, 20 Nov 2007 19:01:38 +0100
> From: [EMAIL PROTECTED]
> To: [EMAIL PROTECTED]
> Subject: Re: Bijections (was OM = SIGMA1)
>
>
> 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
>

There is no such ordinal as "omega-1" in conventional mathematics. Keep in mind
that ordinals are always defined as sets of previous ordinals, with 0 usually
defined as the empty set {}...So,

0 = {}
1 = {0} = {{}}
2 = {0, 1} = {{}, {{}}}
3 = {0, 1, 2} = {{}, {{}}, {{}, {{}}}}

...and so forth. In thes terms, the ordinal "omega" is the set of finite
ordinals, or:

omega = {0, 1, 2, 3, 4, ... } = too much trouble for me to write out in brackets

How would the set "omega-1" be defined? It doesn't make sense unless you
believe in a "last finite ordinal", which of course a non-ultrafinitist will
not believe in.

Jesse

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