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Topic: Countably Infinite Sets
Replies: 2   Last Post: Dec 28, 2012 12:04 AM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Countably Infinite Sets
Posted: Dec 28, 2012 12:04 AM
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On Dec 27, 10:37 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> Butch Malahide <fred.gal...@gmail.com> writes:
> > On Dec 27, 8:43 pm, netzweltler <reinhard_fisc...@arcor.de> wrote:
> >> Does the set {{1}, {1,2}, {1,2,3}, ...} contain all natural numbers?
>
> > No, because it does not contain the natural number 1; if it did, then
> > 1 would be an element of itself, contradiction the axiom of regularity.

>
>   We don't need foundation here. It's a nice exercise in pointless
> pedantry to show there is no representation of the naturals (in ordinary
> set theory) on which the set {{1}, {1,2}, {1,2,3}, ...} contains all
> natural numbers.


Really? How do you prove that? Why can't you have n = {1,2,...,n} for
each natural n?



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