Virgil
Posts:
3,521
Registered:
12/6/04
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Re: Problem with Cantor's diagonal argument
Posted:
Feb 15, 2002 7:08 PM
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In article <a4k5a2$tlnc$1@ID-114100.news.dfncis.de>,
"Harlan Messinger" <hmessinger@erols.com> wrote:
> "Virgil" <vmhjr2@attbi.com> wrote in message
> news://vmhjr2-35D8AB.16133915022002@netnews.attbi.com...
> > Jon Miller <jonathanemiller@home.com> writes:
> >
> > > There are even (intelligent) people who (claim that they) are unable
> > > to believe in infinite sets at all. To them, Cantor's argument
> > > makes no sense.
> >
> > Why is that? Cantor's argument does not presuppose the existence
> > of any infinite set.
> >
> > > There are people who deny the law of the excluded middle. To them,
> proving
> > > that not-A is false does not prove that A is true.
> >
> > There are people who will deny that (P or (not P)) is necessarily
> > true, but those same people will insist that (P and (not P)) is
> > necessarily false. It is the second form which is relevant to
> > Cantor's proof, not the first.
>
> I don't get it. In symbolic logic, isn't "not" *defined* such that (P or
> (not P)) is necessarily true and (P and (not P)) is necessarily false?
>
For most people, yes, but there are those that insist that the
falseness of P is insufficient to guarantee the truth of not P and
vice versa. They deny what is generally known as the law of the
excluded middle, and say that there may be something in the middle,
"between" P and not P.
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