Re: Attempts to Refute Cantor's Uncountability Proof?
On Sat, 08 Jul 2006 04:06:56 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:
I'm interested to know what attempts have been made to refute Cantor's proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from which
one might attempt to refute his argument.
Guffaw.
But before I spend a lot to time
trying to formulate my own argument, it seems reasonable to seek prior art.
Can anybody suggest a source which examines this topic?
The only such source I'm aware of is sci.math. The refutations you
find here make exactly as much sense as someone "refuting" the
construction of the reals from the rationals via dedekind cuts
by pointing out that pi is not rational. Ie, they exhibit a
basic misunderstanding of the argument that they puport to refute.
************************
David C. Ullrich
.
Relevant Pages
- Re: Computable functions/reasls: followup.
... The computable-function definition above still applies, ... Russian-style constructivism is BISH + MP, ... which were reals that you couldn't tell whether or not were rational. ... special about the rationals; they could be replaced by the integers, ...
(sci.logic) - Re: Cantor Confusion
... The "number" pi is definitely a merely fictitious element of continuum. ... naturals, integers, rationals, irrationals, or reals. ... intergers and naturals are genuine. ... genuine numbers to the reals is tempting but not justified. ...
(sci.math) - Re: Dedekind Cuts, Fundamental Sequences: why?
... sequences or Dedekind cuts are useful in defining completeness. ... The constructions mentioned do not succeed in producing a complete ... view of what sort of reasoning about reals, functions of reals, and so ... The motivation for completing the rationals lies mainly in analysis, ...
(sci.math) - Re: PL/I, COBOL, Advantages, Equivalence, et al
... The rationals and the integers have the same cardinality - they can be placed in 1-1 correspondence, but that does not make them isomorphic. ... the rationals are also a ring because after all a field is a ring with additional properties. ... There are several methods for constructing the reals from the rationals. ... In other words, all roots of polynomial equations with either real or complex coefficients are complex numbers and every such equation has a complete set of n roots, where n is the degree of the equation (the fundamental theorem of algebra). ...
(comp.lang.pl1) - Re: PL/I, COBOL, Advantages, Equivalence, et al
... The rationals and the integers have the same cardinality - they can be placed in 1-1 correspondence, but that does not make them isomorphic. ... Algebraically, the integers are a ring, in fact an integral domain, and the rationals are a field of characteristic 0. ... There are several methods for constructing the reals from the rationals. ... In other words, all roots of polynomial equations with either real or complex coefficients are complex numbers and every such equation has a complete set of n roots, where n is the degree of the equation (the fundamental theorem of algebra). ...
(comp.lang.pl1) |
|
