
Re: Countably Infinite Sets
Posted:
Dec 27, 2012 10:23 PM


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.

