Uirgil
Posts:
184
Registered:
4/18/12


Re: Cantor's argument and the Potential Infinite.
Posted:
Nov 17, 2012 12:42 PM


In article <k87e41$nr0$1@dontemail.me>, "LudovicoVan" <julio@diegidio.name> wrote:
> "Uirgil" <uirgil@uirgil.ur> wrote in message > news:uirgilC5CD34.15035616112012@BIGNEWS.USENETMONSTER.COM... > > In article <k851n8$fgs$1@dontemail.me>, > > "LudovicoVan" <julio@diegidio.name> wrote: > >> "Uirgil" <uirgil@uirgil.ur> wrote in message > >> news:uirgil8D50A0.02310116112012@BIGNEWS.USENETMONSTER.COM... > >> > In article <k850hm$a03$2@dontemail.me>, > >> > "LudovicoVan" <julio@diegidio.name> wrote: > >> >> "Uirgil" <uirgil@uirgil.ur> wrote in message > >> >> news:uirgil981B6A.02055216112012@BIGNEWS.USENETMONSTER.COM... > >> <snipped> > >> > >> >> > ZFC offers a standard set theory in which actually infinite sets are > >> >> > not > >> >> > only allowed but actually required to exist, and no one yet has been > >> >> > able to show that ZFC is not a perfectly sound set theory. > >> >> > >> >> That is only because you are so incoherent as to insist to call N an > >> >> actual > >> >> infinity. > >> > > >> > In ZFC, the N is an actually infinite set. So until you can show that > >> > ZFC is internally inconsistent, which no one has yet done, we have > >> > actual infinities in ZFC. > >> > >> That's interesting: would you be so kind to show me how/why, technically > >> although informal as it needs be, N is an "actual infinity" in ZFC? > > > > ZFC requires the existence of a set N such that > > {} is a member of N, and > > If x is a member of N, so is x \/ {x}, and > > N is a subset of every set S such that > > {} is a member of S and > > If x is a member of S, so is x \/ {x} > > > > Such a set is provably not finite, as finiteness of a set would require > > that it biject with some MEMBER of such an N, which N provably does not. > > Sure, N is the minimal set with 0 and closed under the successor operation. > > But that remains a characterization of a *potential infinity*. > In ZFC that particular set is actual.

