Re: Attempts to Refute Cantor's Uncountability Proof?
- From: "david petry" <david_lawrence_petry@xxxxxxxxx>
- Date: 9 Jul 2006 14:55:09 -0700
Stephen Montgomery-Smith wrote:
... if you reject the infinity upon infinities that modern set theory
postulates, then the truth or falsity of Cantor's result becomes
meaningless.
Well said. I believe that is the point most of the "anti-Cantorians"
are trying, but failing, to make.
Nevertheless the logic behind his argument appears in many other places.
For example, you can rewrite the diagonal argument to create a
constructive proof that transcendental numbers exist,
That's not really true. In order to give a constructive proof of the
existence of transcendental numbers, you must show that the digits of
algebraic numbers can actually be computed (and hence, there are bounds
on how closely such numbers can be approximated by rationals), but once
you have shown that, you don't really need the diagonal argument to
show the existence of real numbers that are not algebraic.
.
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