Obections to Cantor's Theory (Wikipedia article)
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From: Daryl McCullough on 26 Jul 2005 15:17 Tony Orlow (aeo6) wrote: > >imaginatorium(a)despammed.com said: >> Remind us how you determine the bigulosity of the set >> { 1, 1/2, 1/4, 1/8,...} >Okay, I don't know what I was saying. It relies on bijections, but doesn't >consider any bijection to mean equality. The mapping functions are used to >determine relative size when it comes to numeric sets. Sorry. So, you agree that for *finite* sets, two sets have the same bigulosity if and only if there is a bijection between the two? But that no longer holds for infinite sets? Then how is bigulosity an improvement over cardinality? -- Daryl McCullough Ithaca, NY
From: Virgil on 26 Jul 2005 15:44 In article <MPG.1d50173fe4d9949d989f8a(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > It seems like axioms are given a status that makes > them unquestionable and almost incomprehensible, beyond the > application of them. For instance, everyone's dismissal of the > infinity inherent in the recursive nature of inductive proof is a > sign that the axiom is not really understood, but accepted without > question. For me, mathematics without meaning is unsatisfying, and > symbolic manipulation without understanding is boring. With a closer > examination of axioms and a willingness to revise them, I would hope > that mathematicians could develop a fully integrated set of universal > axioms which would cover all of math. But, that won't happen as long > as axioms are taken without question, just because they "work". TO again conflates the issues. The choice of axioms to make up an axiom system is one thing, and the composition of the axiom systems that TO finds so barren are actually the result of an extremely long and arduous series of trial and error refinings of a complexity well above TO's capacity for understanding. Once an axiom system is chosen, one must abide by those axioms if that system is to be given a fair evealuation. Since TO will not abide by the axioms of any system containig the equivalent of the Peano axioms, all his claims must refer to what occurs in some other system on his own invention, for which he has not, and perhaps cannot, give a comprehensive axiom system.
From: Virgil on 26 Jul 2005 16:04 In article <MPG.1d5028071ff97742989f8f(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Actually there are two ways to look at it. In unsigned binary, yes, an > infinite > number of 1's is the largest number possible. Since we start with all 0's > representing 0, the size of the set, N, will be one more than 111...111. It > will be 000...001:000...000, or one unit infinity. Until TO can produce unambiguous rules comparing sizes for all of these alleged pseudostrings of binary digits, it is all garbage. Unless TO wants to say that his "naturals" are not naturally ordered and not orderable. For example, given 1000...010...001 and 100...0110...001, where each ellipsis represents a psuedostring of infinitely many zeros, by what rule does one compare the composite psuedostrings to determine which is larger? Note that were the ellipses to each represent only finitely many zeros, the answer would be relatively trivial. And what is the effect of concatenation of two or more such allegedly infinite psuedostrings?
From: Virgil on 26 Jul 2005 16:12 In article <MPG.1d5037fa489fd6b4989f90(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Robert Kolker said: > > Han de Bruijn wrote: > > > > > > True. That's why: > > > > > > A little bit of Physics would be NO Idleness in Mathematics > > > > You are pissing and moaning that a peach is not a pear. Mathematics > > is deductive. Physics is empirical. Mathmematics is about the > > relation of ideas. Physicis is about how the real world works. Two > > completely different things. > > > > Bob Kolker > > > Mathematics is NOT purely deductive. Before you can apply your rules > to the facts at hand, you have to develop the rules. Without an > inductive process of establishing axioms, you have no tools with > which to deduce anything. So, dismissing the question of whether > certain axioms are correct, and ignoring inductive arguments > concerning them, is a real mistake that occurs too often in > mathematics. It's largely at the root of this particular problem. If TO thinks that Peano axioms, ZF, ZFC and NBG were just dreamed up to make his life miserable, he is totally ignorant of the history of mathematical development over the last century or two. TO would benefit by learning a little about how these axiom systems were developed before suggesting that there was no inductive process involved. If ignorance truly were bliss in this matter, TO would be continually estatic.
From: Tony Orlow on 26 Jul 2005 16:16
Robert Low said: > Tony Orlow (aeo6) wrote: > > The problem comes whenever someone wants to use the word "infinite" in any > > other context. > > The problem comes when people use a word in one context as if > its meaning were specified by a different context. That's what > an amphiboly is. > > > There ARE no infinite numbers, only infinite sets, in your > > lexicon. > > Not only do we know about infinite numbers, but there are > at least two different kinds of them: cardinal and ordinal ones. > (Still other notions of infinite number occur in other contexts.) > Again, clear definition of what is meant in a particular > context is required. > > > Talk about needing to learn about language. What is so hard about > > infinite numbers, or trying to define the word itself, independent of bijection > > genuflection? > > What is so hard about the idea of context sensitivity, and accepting > that the word 'infinite' can mean different things in different > contexts? Does it bother you than 'bread' can refer to products > made from different grains, or can even refer to 'money', depending > on the context? > So, you think there can be infinite whole numbers, but that they are not in the infinite set of whole numbers? Why am I being challenged on the very notion of such numbers, if their existence is already established? -- Smiles, Tony |