Obections to Cantor's Theory (Wikipedia article)
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From: Torkel Franzen on 22 Jul 2005 10:35 Barb Knox <see(a)sig.below> writes: > At the risk of appearing to give aid and comfort to the crackpots, let > me make the pedantic point that non-standard models of the (1st-order) > Peano axioms DO contain numbers which are infinite. To say that non-standard elements are "infinite" is only to say that they satisfy a certain infinite set of formulas.
From: Daryl McCullough on 22 Jul 2005 10:38 Han de Bruijn says... >No contradiction. That's perhaps the only good thing about it [set theory]. >If that is the only thing you care about, let me tell you that most of >us care about other things, such as a physics that has a reliable >mathematical machinery at its disposal. Give an example of a calculation in physics in which using modern ("Cantorian") mathematics gives you the wrong answer, and using some other kind of mathematics gives you the right answer. -- Daryl McCullough Ithaca, NY
From: malbrain on 22 Jul 2005 17:28 David Kastrup wrote: > 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. Axioms are agreements -- an shared expression of faith. > 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. What does this mean, it seems backward? You deny one axiom in favor of agreement with another. > > Axioms are posits, or assumptions. They are NOT self evidence > > truths. > > They are not truths at all, they are tools. > Axioms are agreements to share pre-conceived truths. Yes, they are also tools. karl m
From: david petry on 22 Jul 2005 18:03 Han de Bruijn wrote: > 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 hit's the nail on the head. Thanks Han. For those who missed it, the key sentence is: "But, at least, there should exist a path from the abstractions back to the observable objects."
From: David Kastrup on 22 Jul 2005 19:29
malbrain(a)yahoo.com writes: > David Kastrup wrote: >> 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. > > Axioms are agreements -- an shared expression of faith. Uh, no. Axioms have nothing to with faith at all. If you are playing chess, you don't have _faith_ that a knight moves always two squares and then one perpendicular. If it moves differently, that does not cause you to lose faith in the knight, but rather in your opponent's mental sanity. Axioms are the rules of the game. They are arbitrary, but it does not usually make sense to question them, since the purpose is to _play_ the game. Only when you are intend to design a new game does it make sense playing with the rules. >> 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. > > What does this mean, it seems backward? You deny one axiom in favor > of agreement with another. I don't deny it. I just decide I want to play a game with different rules. That does not make the rules for the original game less valid. > Axioms are agreements to share pre-conceived truths. Yes, they are > also tools. Then the movements of pieces in chess are also pre-conceived truths. I find this an odd way of looking at them. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |