Obections to Cantor's Theory (Wikipedia article)
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From: Ross A. Finlayson on 23 Jul 2005 19:37 Jesse F. Hughes wrote: > "david petry" <david_lawrence_petry(a)yahoo.com> writes: > > > 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." > > Seems like nonsense to me. > > You are still claiming that an abstraction is useful, but applying > deductive logic to the abstraction may yield false results unless we > check nature. But this has two obvious problems: > > (1) The abstraction doesn't exist in nature. Now, maybe that means > (to you) that there is no "path from the abstractions back to the > observable objects", but then why did you call infinity a *useful* > abstraction? Or what path do you have in mind and how does it > contradict Cantor's theorem? > > (2) How can any useful abstraction fail to satisfy the basic laws of > deductive logic? You have still not explained that. If our useful > abstraction justifies the presumption of A and A -> B, then it damn > well better justify the presumption of B and there's no need to check > nature to see. > > But that is all that Cantor's theory uses, too. Basic axioms prove a > perfectly simple result. You agree that the proof is correct, but > want to claim that normal logic doesn't apply to infinite sets (which > are nonetheless "useful". > > Of course, you don't give any principle to determine *what* > derivations involving infinite sets are acceptable, aside from "look > out the window and see how infinite sets really behave." > > Your position is really incoherent. You can't simply say that > infinite sets are legitimate mathematical objects but that normal > logic doesn't apply to them. You must have a principled objection > (not "Cantor violates my intuitions." or "Everything before Cantor was > about computation.") and you must have a solution. Which axioms are > wrong? Do you really suppose that deductive logic is not the right > logic for set theory? Then what is? > > Anyway, viva la revolution and all that. > > -- > "No feeling sympathy for mathematicians who start marching with signs > like 'Will work for food' in the future... I will not show mercy > going forward. I was trained as a soldier in the United States Army > after all... We play to win." --James Harris, feel his wrath! Now, that's a pearl. Ross -- "Anyway, viva la revolution (sic) and all that." - Jesse F. Hughes
From: David Kastrup on 23 Jul 2005 19:47 "david petry" <david_lawrence_petry(a)yahoo.com> writes: > My claim is that all the results of analysis which have observable > implications, an hence have the potential to be applied as models of > the real world, necessarily do not need Cantor's Theory. So what? There is no shortage of unsubstantiated claims, so this is hardly Earth-shattering material. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Martin Shobe on 23 Jul 2005 19:49 On 23 Jul 2005 14:32:08 -0700, Han.deBruijn(a)DTO.TUDelft.NL wrote: >An idealization _never_ exists in nature. As far as the idealization >called 'infinity' is concerned, the classical concept of a limit has >indeed a path back to earth, called: inaccuracy, errors, sloppyness. >The gist is that the delta's and the epsilon's in the limit concept >are idealizations from real measurement "errors". All we have to do >with materialization (the way back) is to re-introduce these errors >and accept them as "real". Or if N -> oo, replace it by "very big". We don't use sequences and limits to deal with errors. We use probability or intervals. >> But that is all that Cantor's theory uses, too. Basic axioms prove a >> perfectly simple result. You agree that the proof is correct, but >> want to claim that normal logic doesn't apply to infinite sets (which >> are nonetheless "useful". > >The problem is with the "basic axioms", not with the logic. These >axioms >are definitely NOT a proper idealization from what physicists would >call >a "set" into what Cantor has defined as a set. See for example: > > http://huizen.dto.tudelft.nl/deBruijn/grondig/science.htm#cm Of course, there is no need for it to be so. There is no need for "set" as a physicist would understand it to have anything to do "set" as a mathematician would understand it. Martin
From: Dik T. Winter on 23 Jul 2005 19:58 In article <1122150386.652953.119310(a)g43g2000cwa.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > Daryl McCullough wrote: > > 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. > > That's actually not difficult. Many mainstream mathematicians find > that physicists are quite sloppy when they are doing calculus: they > blindly assume, for example, that every function which is _defined_ > on the reals is automatically continuous there as well. This is not > a valid assumption within common mathematics. But, in Intuitionism > this result is true, and known as Brouwer's Continuity Theorem: It is true in Brouwer's mathematics because he has a different view on the reals than mainstream mathematics. In Brouwer's mathematics the "function" f(x) = 0 if x >= 0 and f(x) = 1 if x < 0 is not a function. That is, you can not split the reals in two non-empty sets A and B such that A union B = R and A intersection B = 0. So in his view the function f either is not defined for some arguments, or it has two defined values for some arguments. (I think he would opt for the first because he rejected the principle of the excluded middle.) And indeed, there is a part of current mathematics that also does reject the PEM, and finds that every function is differentiable, but does so on quite different grounds. (In their view the above function *is* a function, but is differentiable.) But that is also not mainstream. But have a look at: <http://math.ucr.edu/home/baez/topos.html>, which explains a few things much better than I could do. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: MoeBlee on 23 Jul 2005 21:09
Han de Bruijn, My message here is hardly as monumental as that given in Russell's letter to Frege, but at http://huizen.dto.tudelft.nl/deBruijn/grondig/science.htm#cm you stated that it is clear that the theory you propose is consistent with what you call the "finitistic part of ZFC". Not only is that not clear; it is incorrect. Your own axiom along with just basic Z set theory allows the following two formulas as theorems: Ex x e 0 ~ Ex x e 0 where '0' stands for the empty set. MoeBlee |