Obections to Cantor's Theory (Wikipedia article)
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From: Helene.Boucher on 22 Jul 2005 03:41 Jeffrey Ketland wrote: <snipping a fine exposition of Hajek and Pudlak> Thanks for that. On the other hand my original concern is what happens when the theory does not have (as Q does) the axiom "All non-zero numbers have a predecessor." This is what I meant by "PA - induction," since definitions of PA without reference to Q do not include this axiom (since it would be redundant, since it can be proven by induction). My apologies for misreplying to your query about what I meant by "PA - induction." It's led to a lot of ink under the bridge!
From: Han de Bruijn on 22 Jul 2005 03:51 stephen(a)nomail.com wrote: > In sci.math Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>stephen(a)nomail.com wrote: > >>>Why are you bringing physics into this? Whether black >>>holes exist or not has nothing to do with set theory. > >>Theoretically: no. In practice: yes. Because some consequences of set >>theory have invaded into physics by the fact that mathematics becomes >>somewhat _applied_ there, huh ! Geez ... > > Are you claiming that set theory is directly applied in General > Relativity? Do you actually have evidence of that? Ignoring > mathematical foundations, which physicists, and most everybody else, > typically do, it seems that the limit of 1/sqrt(1-v^2/c^2) as > v approaches c is infinite with or without set theory. This is the usual argument that foundational issues in mathematics "can do no harm" when it comes to applications outside. I do not agree with this argument, having gathered too much evidence of the contrary. Now I don't say that mathematicians should be blamed for this, if it happens. But the fact is that most physicists have a blind faith in mathematics and can be easily deluded by the fact that infinities actually exists, within mainstream mathematics, and think that they do exist in physics as well. Yes, I have evidence. Read this: http://huizen.dto.tudelft.nl/deBruijn/grondig/science.htm#cz A "naked singularity" was found in a heat exchanger. But since a naked singularity cannot possibly exist there, it leads to quite a different conclusion: http://hdebruijn.soo.dto.tudelft.nl/jaar2004/IHXTAK.pdf Han de Bruijn
From: Han de Bruijn on 22 Jul 2005 03:55 Robert Low wrote: > stephen(a)nomail.com wrote: > >> Are you claiming that set theory is directly applied in General >> Relativity? Do you actually have evidence of that? > > I have seen a paper in which transfinite induction was > used. And in the existence of maximal solutions to the > initial value problem, appeal is (at least sometimes) > made to Zorn's lemma. I'm not surprised. Whether people like it or not, mathematics cannot be separated from its applications. Han de Bruijn
From: David Kastrup on 22 Jul 2005 04:10 Robert Kolker <nowhere(a)nowhere.com> writes: > Peter Webb wrote: >> How do you disagree with an axiom? > > By assuming a contrary axiom, as is done in non-Euclidean > geometry. Oh, but that is not disagreeing with it. In fact, it is expressing faith that the axiom indeed _is_ an axiom. If I use a hammer for driving a nail, this does not mean that I disagree with a screwdriver. I'll still use the screwdriver when having to drive a screw. > The parallel postulate is denied in non-euclidean geometry. It is not as much denied as exchanged. > Axioms are posits, or assumptions. They are NOT self evidence > truths. They are not truths at all, they are tools. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Han de Bruijn on 22 Jul 2005 04:16
Jesse F. Hughes wrote: > Certainly infinite sets and power sets exist as absractions. But, > abstractions don't necessarily obey exactly that same laws of logic > as directly observable objects. Infinite sets exist as (improper) idealizations. Indeed, idealizations don't necessarily obey the same laws of logic as the directly observable objects. But, at least, there should exist a path from the abstractions back to the observable objects. Directly observable sets are idealized - and that's a good thing - but idealized sets must also be "materialized" again, for the sake of _applications_. The latter tends to be forgotten. That's basically what frustrates the Applied and causes anti-Cantorism. Han de Bruijn |