Obections to Cantor's Theory (Wikipedia article)
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From: Jiri Lebl on 21 Jul 2005 12:43 david petry wrote: > I'm in the process of writing an article about > objections to Cantor's Theory, which I plan to contribute > to the Wikipedia. I would be interested in having > some intelligent feedback. Here' the article so far. I would avoid absolute claims in the article such as "pure mathematicians think this" etc... > While the pure mathematicians almost unanimously accept > Cantor's Theory (with the exception of a small group of > constructivists), there are lots of intelligent people who > believe it to be an absurdity. Typically, these people > are non-experts in pure mathematics, but they are people > who have who have found mathematics to be of great practical > value in science and technology, and who like to view > mathematics itself as a science. Many mathematicians, including pure mathematicians, view mathematics as a science (hence for example: MSRI = Mathematical SCIENCES Research Institute). Further vast majority of applied mathematicians accept Cantor's theory. In fact I would argue that more applied mathematicians probably do then pure mathematicians. This is because of the multitude of useful results that it has brought which are used all over in physics, engineering, statistics and other sciences in general. > These "anti-Cantorians" see an underlying reality to > mathematics, namely, computation. They tend to accept the > idea that the computer can be thought of as a microscope > into the world of computation, and mathematics is the > science which studies the phenomena observed through that > microscope. They claim that that paradigm includes all > of the mathematics which has the potential to be applied to > the task of understanding phenomena in the real world (e.g. > in science and engineering). Tell that to the CS masters student at a local university here that got the unfortunate, purely practical, task of writing a program to detect infinite loops in pascal programs (the idea was to test freshmen programs before running them so that they wouldn't hog the processors). Only after a year of effort did he find out that such a task was in fact impossible. By a theory very similar to Cantor (actually results for computability were derived from Cantor's argument). Most definately not a fantasy world assignment. > Cantor's Theory, if taken seriously, would lead us to believe > that while the collection of all objects in the world of > computation is a countable set, and while the collection of all > identifiable abstractions derived from the world of computation > is a countable set, there nevertheless "exist" uncountable sets, > implying (again, according to Cantor's logic) the "existence" > of a super-infinite fantasy world having no connection to the > underlying reality of mathematics. The anti-Cantorians see > such a belief as an absurdity (in the sense of being > disconnected from reality, rather than merely counter-intuitive). This paragraph seems to be biased towards anti-Cantorians rather then describing what they believe. Given all the results that modern analysis (which is heavily based on Cantor's set theory) has brought us, it seems that reality at least seems to work as in our models which do use Cantor's ideas, rather then not. What we know of the real world is based on models that were derived or proved from Cantor's theory. There is a difference in arguing that Cantor's theory has no basis in reality on intuitive means and in arguing that Cantor's theory is wrong. Perhaps it is that Cantor's theory is only useful in approximate models of the reality. So far no one has come up with a system of mathematics capable of solving real world problems that for example doesn't use the completeness of the real numbers which is the property that guarantees uncountability of the real numbers. In any case, most anti-Cantorians that post to sci.math seem to be not claiming that Cantor's theory is the wrong basis for modeling reality on, but they seem to be claiming that Cantor's theory is wrong mathematically, and that is simply rediculous. > In the contemporary mainstream mathematical literature, there > is almost no debate over the validity of Cantor's Theory. That is because mathematically it is a sound theory. You will NOT find any further debate on this in mainstream mathematical literature. Pure mathematicians are interested in showing something is or is not mathematically correct. Cantor's theory IS correct mathematically, and that is why there won't be any debate over it in mathematical journals. Maybe someone will find a more palatable basis for mathematics which allows the sort of analysis that is needed in other sciences. That won't invalidate Cantor's set theory, and even if such a new theory will be gotten there won't be any debate over Cantor's set theory being wrong in any mathematical journal. > It was the advent of the internet which revealed just how > prevalent the anti-Cantorian view still is; there seems to be a > never-ending heated debate about Cantor's Theory in the Usenet > newsgroups sci.math and sci.logic. Typically, the > anti-Cantorians accuse the pure mathematicians of living in a > dream world, and the mathematicians respond by accusing the > anti-Cantorians of being imbeciles, idiots and crackpots. All the posters I saw were trying to purport mathematical "proofs" that Cantor's set theory (mostly the diagonalization argument) is wrong. That is the same as people claiming to have squared the circle or claiming to have computed pi to be 3.125. That is why they are called "imbeciles, idiots and crackpots." > It is plausible that in the future, mathematics will be split > into two disciplines - scientific mathematics (i.e. the science > of phenomena observable in the world of computation), and > philosophical mathematics, wherein Cantor's Theory is > merely one of the many possible "theories" of the infinite. This paragraph is total nonsense put in purely because you were lacking another paragraph to put into the article. Mathematics already ranges from mathematical physics, all the way to pure mathematics. It is people on the right end of the spectrum (pure mathematics) that contain those that try to do mathematics without certain tools such as Cantor's set theory. The people on the left, who are more interested in modeling the real world are probably rarely interested if their "model" uses the complete reals or not, they use the best theory suited for the job, and that theory is based on Cantor's set theory. Those who try to find alternatives to Cantor's set theory are interested in the foundations of mathematics, which is about as pure mathematics as you can get. Jiri
From: Chris Menzel on 21 Jul 2005 12:46 On 20 Jul 2005 23:34:33 -0700, imaginatorium(a)despammed.com <imaginatorium(a)despammed.com> said: > David Kastrup wrote: >> "infinitely large" would be a property associated with a single >> member. "arbitrarily large" is a property associated with a >> collection of members that form an infinite subset. > > Seems harsh to niggle with what is a pretty clear exposition of the > basic problem, but you have to be ultra-careful with words hereabouts. > The "arbitrarily large" is a property not of the *collection of > elements*, but rather a property of the individual elements, under > some quantification - I guess I don't see how "arbitrarily large" could be a property of any individual number, and I don't understand what "under some quantification" adds -- though that might well be simply a failure of imagination on my part. At the same time, it doesn't seem entirely natural to say that being arbitrarily large is a property of a *set* of numbers either. Natural language (English, anyway) suggests its a property of the elements, collectively, like the property of, say, lifting a piano, applied collectively to four men: The members of S are arbitrarily large iff, for each n in S, there is an m in S that is larger than n. Granted, if you think plurals like "the men" or "the members of S" refer to sets, then the property in question is a property of sets, but that's at the least a questionable view of the semantics of plurals. (The *set* of four men, for example, didn't lift the piano; the men did.) Chris Menzel
From: Helene.Boucher on 21 Jul 2005 13:08 Jeffrey Ketland wrote: > >> If by "PA minus the induction scheme", you mean the six usual axioms for > >> successor, + and x, > > > > Yes that's what I (and I think most other people) mean. > > I agree. It's usually something like Q (with the definition of <), plus the > induction scheme. > > So, your original questions are, I think, these: Actually it wasn't a question, more a comment... > (a) Does Q + induction scheme imply the least-number principle? > (b) Does Q + induction scheme imply the principle of total induction? > The answer is yes. (See Hajek/Pudlak 1993, _Metamathematics of First-Order > Arthmetic_, p. 35). I wasn't mentioning this direction... > > Or did you mean to ask about the converse in (a), > (c) Does Q + least number principle imply induction scheme? > I'm guessing, since I haven't checked the proof, but I'd be surprised if > that wasn't true as well. > This direction is the problem. See the link I provided (or Google as I suggested).
From: Jesse F. Hughes on 21 Jul 2005 13:08 "david petry" <david_lawrence_petry(a)yahoo.com> writes: >> There is no mention of one historical or living figure who is >> anti-Cantorian, what their objections were > > Hmm, not quite. I did mention Kronecker. Did you miss that? > Since you are intent on expressing common anti-Cantorian ideas, I suppose that you should mention that many anti-Cantorians believe that there are infinite natural numbers. We have seen that claim come up repeatedly in this group (most recently with Tony Orlow). Will you include this viewpoint in your survey of "anti-Cantorians"? 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? -- Jesse F. Hughes "That's what's brutal about mathematics! When you're wrong, you can have spent years, and lots of effort, and come out at the end with nothing." -- James S. Harris on the path of self-discovery (?)
From: Jeffrey Ketland on 21 Jul 2005 14:10
Helene.Boucher(a)wanadoo.fr >> Or did you mean to ask about the converse in (a), >> (c) Does Q + least number principle imply induction scheme? >> I'm guessing, since I haven't checked the proof, but I'd be surprised if >> that wasn't true as well. >> > > This direction is the problem. See the link I provided (or Google as I > suggested). I looked at the link. But it appears to concern the equivalence of PHP and induction, a different topic (though the topic of these equivalences is discussed in Hajek/Pudlak). We have that PA- + least number principle implies induction, where PA- is the base theory Kaye describes. Also, the least number principle and order-induction are equivalent (in predicate logic). So: the question is: does Q + order-induction imply induction? This is how Hajek and Pudlak discuss the topic. They discuss this for classes of Sigma_n and Pi_n formula. Let L_{phi} be the least-number principle formula for phi. Let I_{phi} be the induction formula for phi. Let I'_{phi} be the order-induction formula for phi. The theory ISigma_n is Q U {I_{phi} : phi a Sigma_n formula}. The theory I'Sigma_n is Q' U {I'_{phi} : phi a Sigma_n formula}. where Q' is Q augmented with the axiom x < Sx. Lemma 2.12 (p. 64) of Hajek/Pudlak shows that ISigma_n is equivalent to I'Sigma_n, and IPi_n is equivalent to I'Pi_n. They remark (p. 63) that "it is unknown whether IDelta_n and LDelta_n are equivalent" (although LDelta_{n+1} implies IDelta_{n+1}.). Hajek and Pudlak are mainly concerned with fragments of PA defined by the complexity of formulas. I can't see if they show that Q + order-induction implies ordinary induction, but I shall think about it. Perhaps it's an open issue. Finally, the axioms of PA- are all theorems of ISigma_1 (possibly even IOpen). Therefore, we have ISigma_1 + order-induction implies ordinary induction. What the small amount of ordinary induction is used for is to show the general algebraic properties of + and x (commutativity, etc.) and the general order properties of < (i.e., discrete linear order with a least element). In short, order-induction (equiv., least number principle) implies ordinary induction modulo a weak base theory which includes these properties of +, x and <. --- Jeff |