Re: GCH vs. Axiom of Choice.
From: Michael Stemper (mstemper_at_siemens-emis.com)
Date: 12/14/04
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Date: Tue, 14 Dec 2004 12:23:28 -0600
In article <20041211214827.14166.00000243@mb-m13.aol.com>, KRamsay writes:
>In article <1102609116.767364.64310@f14g2000cwb.googlegroups.com>, "J.E." <troubled6man@yahoo.com> writes:
>| so I didn't think "C" or "~C" was
>|considered "false", just relatively independant axioms.
>
>Are you assuming that every statement X for which we can prove
>Con(ZF)=>Con(ZF+X) and Con(ZF)=>Con(ZF+~X) is neither true
>nor false? I don't mean it as a rhetorical question.
At the risk of sounding like Pilate, "What is truth?" I always thought
that "Mathematical Truth" was only definable with respect to a given
set of axioms.
I guess that I got this idea from (popularized) readings about the
Parallel Postulate. Some guy spends his life trying to prove it by
reductio ad absurdum and fails. People build on his work, giving us
new forms of geometry. Each of these forms of geometry is, to the
best of my knowledge, considered "true". (Obviously, you can't take
the results of one and apply them with another.)
>I don't assume anything like that, and to me this seems like a
>nonsequitur. If you believe such a thing, and you don't know
>why you believe it, it's liable to be a source of some confusion.
So, is the Parallel Postulate true all by itself -- whatever that means?
If not, what's different about C?
-- Michael F. Stemper #include <Standard_Disclaimer> Time flies like an arrow. Fruit flies like a banana.
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