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Topic: Cantor's argument and the Potential Infinite.
Replies: 17   Last Post: Nov 17, 2012 10:59 PM

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Uirgil

Posts: 183
Registered: 4/18/12
Re: Cantor's argument and the Potential Infinite.
Posted: Nov 16, 2012 5:03 PM
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In article <k851n8$fgs$1@dont-email.me>,
"LudovicoVan" <julio@diegidio.name> wrote:

> "Uirgil" <uirgil@uirgil.ur> wrote in message
> news:uirgil-8D50A0.02310116112012@BIGNEWS.USENETMONSTER.COM...

> > In article <k850hm$a03$2@dont-email.me>,
> > "LudovicoVan" <julio@diegidio.name> wrote:

> >> "Uirgil" <uirgil@uirgil.ur> wrote in message
> >> news:uirgil-981B6A.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?
>
> -LV
>


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.



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