Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 21 Jul 2005 14:18 In article <84c62$42df5560$82a1e3ad$10805(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > Every bit of "Cantorianism" has been well enough defined for the > > understanding of thousands upon thousands of people. That TO fails where > > so many have succeeded says more about TO than about the adequacy of > > "Cantorianism's" explanations. > > The fact that a faith has millions of adherants doesn't say anything > about its validity. It says something about the society wherein it is > accepted, though. > > Han de Bruijn If TO or WM or anyone else wants to object to the "faith" of "Cantorianism" the way to do it is not to complain about the consequences of the axioms on which they are based but to replace that axiom system with something that is at least not self-contradictory. But that is not what the typical anti-Cantrian does.
From: Jesse F. Hughes on 21 Jul 2005 15:12 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > "david petry" <david_lawrence_petry(a)yahoo.com> writes: > [...] > Also, when *will* you define what you mean by Cantor's theory? Do you > mean ZF? If so, why call it Cantor's theory? If not, what *is* > Cantor's theory? Actually, I see that he made motions towards this in the other thread "Updated:...". But it is still utterly opaque how one is supposed to react to his criticisms. Petry doesn't say that ZF is bad exactly, but that some reasoning about infinite sets should be disallowed. Which reasoning? As far as I can tell: that which yields results he finds absurd. There is no obvious principle to be derived from criticisms like: 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? They're okay. Power set too. But reasoning about those is iffy, according to Petry. They might not obey the same laws of logic as "observable objects" (like what? 2? Pi? The set of primes less than 10^10^10?). Might not? Uh oh. -- "Clouds are always white and the sky is always blue, And houses it doesn't matter what color they are, And ours is made of brick." -- A new song by Quincy P. Hughes
From: Daryl McCullough on 21 Jul 2005 16:29 Han de Bruijn says... > >Virgil wrote: > >> One issue here is that TO keeps ignoring standard mathematical >> definitions, however often presented, and then declaring that that the >> defined words and phrases must have other meanings than the ones >> mathematicians have agreed on. > >Meanings ? Agreed on ? > >Look at Daryl McCullough's arguments, where he asks Tony to think about >"fluffy pink flying elephants". You've got that completely backwards. Tony is the one who wants to talk about undefined terms. I want to stick to basic mathematical terms: sets, naturals, membership, addition, multiplication, functions, etc. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 21 Jul 2005 16:37 Han de Bruijn says... > >Daryl McCullough wrote: > >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >> >>>I asked for a definition of infinite, and no one could give me a >>>definition of that word. The best I could get was that an infinite >>>set can have a bijection with a proper subset, which is hardly a >>>definition of the word "infinite". >> >> On the contrary, that's a perfectly good definition of the concept >> "infinite set". > >It's the standard definition of the "actual infinite", but it is not >"perfectly good". Worse. It's not good at all. It's perfectly good in the sense that it allows for mathematical reasoning about the infinite. We can reason that certain sets must be infinite, and that other sets must be finite. In contrast, Tony's talk about infinite naturals doesn't allow any kind of reasoning to be made about them. -- Daryl McCullough Ithaca, NY
From: G. Frege on 21 Jul 2005 17:07
On 21 Jul 2005 13:37:22 -0700, stevendaryl3016(a)yahoo.com (Daryl McCullough) wrote: > > In contrast, Tony's talk about infinite naturals doesn't allow > any kind of reasoning to be made about them. > I read: "Tony's talk ... doesn't allow [for] any kind of reasoning..." Same for Mýckenheim, btw. F. |