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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 17, 2012 12:42 PM
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In article <k87e41$nr0$1@dont-email.me>,
"LudovicoVan" <julio@diegidio.name> wrote:

> "Uirgil" <uirgil@uirgil.ur> wrote in message
> news:uirgil-C5CD34.15035616112012@BIGNEWS.USENETMONSTER.COM...

> > 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?

> >
> > 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.



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