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Re: Problem with Cantor's diagonal argument
Posted:
Feb 16, 2002 12:36 AM
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"Harlan Messinger" <hmessinger@erols.com> wrote in message news:<a4k5a2$tlnc$1@ID-114100.news.dfncis.de>...
> "Virgil" <vmhjr2@attbi.com> wrote in message
> news://vmhjr2-35D8AB.16133915022002@netnews.attbi.com...
> > Jon Miller <jonathanemiller@home.com> writes:
> >
[snip]
> >
> > > 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?
The propositional calculus can be "fragmented" by taking out certain axioms
like (P or (not P)), the "law of the excluded middle" mentioned above. One
obtains weaker (but therefore consistent) systems of logic this way.
In particular proponents of "constructive" mathematics and specifically
"intuitionist" logic eschew the law of the excluded middle.
Regards,
Chip
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