LudovicoVan
Posts:
2,971
From:
London
Registered:
2/8/08
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Re: Cantor's argument and the Potential Infinite.
Posted:
Nov 17, 2012 2:32 PM
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"Uirgil" <uirgil@uirgil.ur> wrote in message
news:uirgil-4EE436.10423717112012@BIGNEWS.USENETMONSTER.COM...
> 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.
Yet you have shown no ground to call it actual, in fact the opposite.
-LV
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