
Re: Countably Infinite Sets
Posted:
Dec 28, 2012 12:04 AM


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?

