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Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Countably Infinite Sets
Replies: 11   Last Post: Dec 28, 2012 9:08 PM

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Butch Malahide

Posts: 885
Registered: 6/29/05
Re: Countably Infinite Sets
Posted: Dec 27, 2012 10:23 PM
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On Dec 27, 9:08 pm, Butch Malahide <fred.gal...@gmail.com> wrote:
> 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.

Sorry for the typo; I meant, contradictiNG the axiom of regularity.

By the way, I wonder if you meant to ask:
Does the set {0}, {0,1}, {0,1,2}, ...} contain all natural numbers?
It does, if you identify natural numbers with the corresponding Von
Neumann ordinals.

On another note, the set
N union {{1}, {1,2}, {1,2,3}} = {{1}, {1,2}, {1,2,3}, 1, 2, 3, 4,
5, ...}
also contains every natural number.

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