Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 25 Jul 2005 17:55 In article <MPG.1d4efaa1e74d67ef989f74(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > David Kastrup said: > > Helene.Boucher(a)wanadoo.fr writes: > > > > > Daryl McCullough wrote: > > > > > >> Instead of using the term "size" to refer to sets, we could > > >> refer to the "bloppitude". > > >> > > >> Instead of using the words "infinite", we could use the term > > >> "mega-bloppity". > > >> > > >> Nothing of any importance about mathematics would change > > >> if we substituted different words for the basic concepts. > > >> > > > > > > Shouldn't there be one "p" in "blopptitude?" > > > > > > Anyway, nothing in mathematics would change, but surely the interest > > > of the resulting propositions would diminish, should the word "size" > > > disappear. People think (and IMHO in error) that the definition > > > captures correctly the concept "size". > > > > Increase your set size! This all-natural new formula will give you > > the true superset you always dreamt of! No injections required! Now > > you can fill all pigeonholes and still have your member available for > > more! > > > > Hear the testimonials: > > > > WM: It actually has infinite potential. Certainly the largest finite > > possible. > > > > TO: It goes beyond finite. I did things on the Peano that others > > claimed impossible. Gives a whole new meaning to the word. > > > > You are probably right. The interest of the resulting propositions > > would diminish. > > > > > Pretty Funny! At least you have a sense of humor. :) Which puts him one up on TO.
From: Virgil on 25 Jul 2005 18:08 In article <MPG.1d4efc4011f1c632989f75(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow writes: > > > > > >Daryl McCullough said: > > > > >> Once again, if your claims had any merit whatsoever, then you > > >> would be able to rephrase them in a way that does not rely on > > >> unorthodox meanings of terms. Rephrase your claim without using > > >> the word "finite" or "infinite". Is that possible? > > > > >Are you talking to me or Dik? > > > > You. You're the one taking about infinite naturals. > Dik made a statement, which you have now snipped, to which you were > responding, > I believe. > > > > >No one else seems to have a definition for "infinite" as a > > >stand-alone word. > > > > Mathematicians have a definition for "infinite set", > > namely "a set such that there exists a bijection between > > that set and a proper subset of itself". > yeah, I know. > > > > >Finite means with an end > > > > I don't want more words. I want a *mathematical* definition. > > Give a definition in terms of the standard mathematical concepts > > such as "subset", "element of", etc. > > > > >If the leftmost possible significant digit is at position n, > > >then the largest number possible in base b is b^n. > > > > Who says that there is a leftmost possible significant digit? > GIVEN n as finite, b^n is finite. Iff n can be infinite, then b^n can be > infinite. > > > > You are confusing two different things: > > > > (1) For every string, there is a finite number n such that > > the length of the string is less than n. > > > > (2) There is a finite number n such that for every string > > the length of the string is less than n. > > > > (1) is true, but (2) is false. > They are both false if you allow infinite strings. What makes you think i am > confusing those two statements? To avoid the ambiguity on which TO is attempting to weasel out consider: A finite string means one in which the string positions are numbered with a finite set of set finite naturals having a last member. (1) For every finite string, there is a finite number n such that the length of the string is less than n. (2) There is a finite number n such that for every finite string the length of the string is less than n. Or even more clearly: (1) for every natural n, there is a natural m such that m > n. True, since on can always take m = n+1. (2) there is a natural m such that for every natural n, m > n. False, when n = m+1. Can TO cross this pons asinorum? Not if he is still claiming the statements are equivalent.
From: Virgil on 25 Jul 2005 18:18 In article <MPG.1d4f01ca1b87261c989f76(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > hale(a)tulane.edu said: > > > > > > Tony Orlow (aeo6) wrote: > > > malbrain(a)yahoo.com said: > > [cut] > > > > That's not what our axiom says. It says that induction covers > > > > all the natural numbers in a single step, a single > > > > leap-of-faith. > > > That's really not the case. It is a recursive proof where the > > > property is proven true for each element depending on its truth > > > for the preceding element. f(n)->f(n+1), for n=1 to oo. > > > Otherwise, how do you think it proves things for each and every n > > > in N? > > > > Do you agree that one can prove the statement "If n is even, then > > n+1 is odd" without using induction? > > > > I would say that one could. > > > > In such a proof, one would start off with something like: "Suppose > > n is even. Then, there is an m such that n = 2 * n. etc" > > > > In such a proof, you would not be running through elements n = 1 to > > oo. Or, do you disagree with this? If you disagree, then we can > > reduce the problem to this case and eliminate the discussion about > > how induction works. > Are you offering a different type of proof that all whole numbers are > finite? I have seen only the inductive form, so that might be > interesting. > > > > Then, to answer your last question, I prove that f(n) is true for > > each and every n in N by invoking "axiom of induction" after I have > > proved f(1) is true and "f(n)->f(n+1)" is true (this without > > invoking induction). > Okay, so you ARE using induction, invoking it after not invoking it. > Right? > > > > Induction is an axiom, not a recursive proof. The idea of recursing > > from n = 1 to oo is an intuitive justification of the axiom of > > induction. But, an axiom doesn't need a justification (in a certain > > sense). > > > > -- Bill Hale > > > > > Well, Bill, I have come to realize that this is one of my most > central complaints about mathematics as it is today. There is great > emphasis put on axiomatic systems, and axioms are given this status > as unquestionable atomic fact without justification, when really > facts NEED to be justified somehow, from outside of the axiomatic > system. Axioms are taken as fact within an axiomatic system and used > for proofs and arguments, but that doesn't mean every axiom is > universally true as stated, or true at all outside of the system > where they reside. No one says that axioms are true outside of their system, any more that anyone claims that 4 balls and is a walk or three strikes is an out hold outside of baseball. They are true inside of baseball because they are the "axioms" of baseball. No one forces TO to "play the game" but no one will allow TO to change the rules of the game either. If he wants to make up his own game, fine, but he will not find many willing to play by his rules. > In my opinion, TO's opinion is worth no more that his alleged "proofs", that is to say, worth nothing.
From: Virgil on 25 Jul 2005 18:24 In article <MPG.1d4f02a49754dd31989f77(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > >> 1. Phi(0). > > >> 2. for all natural numbers x, Phi(x) implies Phi(x+1). > > > > >The proof has a finite form, much like a recursive algorithm. A recursive > > >algorithm will run forever if it doesn't have some stop condition, like > > >running > > >out of nodes in a tree path, which is bad for a computer program. > > > > But unlike an algorithm, there is no implied infinite number of > > steps. > Yes there is. You show the proprty true for n=0, then for succ(n), then succ > (succ(n)), etc. This is the justification for the axiom. That is why > inductive > proof is said to work. > > > > >Your #2 above is the recursive part of the proof; it proves something > > >true based on the truth of its predecessor. > > > > No, the only thing that you prove is the implication > > Phi(x) implies Phi(x+1). > Yeah that's what i said. > > > > >When you actually "run" the proof, > > > > You don't "run" proofs. > That's why I put it in quotes, but it is really a recursive proof with an > implied loop. > > > > >#2 acts like a loop, iterating its way through all the members > > >of the set, with no stop condition. > > > > No, it's not like that, because you don't "run" proofs. > Why do you think inductive proof is agreed to work? Think. Peano thought. TO didn't. The point that Peano realized, and TO does not, is that if one knows absolutely that one can do the unending recursion, there is no need to. So the inductive axiom says: if you can prove that you can do them all, you don't have to do any of them to get the desired result.
From: Chris Menzel on 25 Jul 2005 18:21
On Mon, 25 Jul 2005 15:27:28 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > Chris Menzel said: >> On Wed, 20 Jul 2005 16:16:44 -0400, Tony Orlow <aeo6(a)cornell.edu> said: >> > Chris Menzel said: >> >> On Wed, 20 Jul 2005 09:08:02 -0400, Tony Orlow <aeo6(a)cornell.edu> said: >> >> > ... >> >> >> To say that a list of the reals is aleph_1 in length assumes the >> >> >> continuum hypothesis; is that what you intend? (And if so, why?) >> >> >> Moreover, granted, aleph_1 is omega_1 in pure set theory, but one should >> >> >> use ordinal numbers rather than cardinals when talking about such things >> >> >> as the length of a list, as there are many lists of different ordinal >> >> >> lengths that are the same size. >> >> >> >> > I don't think it assumes any such thing. >> >> >> >> If the assumption you are referring to is that there is a list of >> >> reals of length omega_1 (= aleph_1 in pure set theory), then all you >> >> are doing by denying that this assumption requires the continuum >> >> hypothesis is showing your abject ignorance of the subject. >> >> > The continuum hypothesis states, if I'm not mistaken, that there are >> > no infinities between aleph_0 and aleph_1. >> >> Sorry, dude, no, you are quite mistaken. aleph_1 is *by definition* the >> next cardinal after aleph_0. The continuum hypothesis is that aleph_1 >> is the size of the set of real numbers. > > Interesting. MathWorld says: > > The proposal originally made by Georg Cantor that there is no > infinite set with a cardinal number between that of the "small" > infinite set of integers aleph_0 and the "large" infinite set of real > numbers c (the "continuum"). Symbolically, the continuum hypothesis > is that aleph_1=c. > > Apparently it says both, dude. No, it really doesn't. The fact that you looked it up suggests you'd genuinely like to understand it, so I'll try to spell it out a little more clearly than the MathWorld article. In arithmetic and real analysis we encounter two particularly interesting examples of infinite sets: the set N of natural numbers and the set R of real numbers. After providing us with a simple and natural way to talk about cardinality and relative size that applies to finite and infinite sets alike, Cantor proved that the size of N is strictly smaller than that of R. So we now have two great examples of distinct *cardinalities* -- that of N, and that of R (often called "c", as above). N's cardinality, aleph_0, is easily shown to be the smallest infinite cardinal number. But what about c? We know it's larger than aleph_0, by Cantor's theorem, but a priori we have no idea how much larger. In particular, we don't know if it is the very *next* infinite cardinal. The continuum hypothesis is that c *is* in fact the next largest infinite cardinal after aleph_0. By definition, we designate the next cardinal after aleph_0 "aleph_1". The continuum *hypothesis*, therefore, is that c = aleph_1. >> Really, you are making such a fool of yourself by talking through your >> hat instead of just learning some simple, basic set theory. Do yourself >> a favor. > > I don't need a favor. Hey, have it your way. >> >> > In fact, it is my position that there is an infinite spectrum of >> >> > infinities between that of the naturals and that of the reals. >> >> Guess what? That means you reject the continuum hypothesis! Isn't that >> exciting? Don't you just want to go out and learn all about it rather >> than just spouting vague, uninformed, and often silly nonsense? > > Why do I want to spend my time studying something that I disagree > with? Why don't you go study Mormonism? Wouldn't that be exciting? If I disagreed with Mormonism and wished to refute its theology (not that I have the least interest in doing so) I'd spend *a lot* of time studying it so I wouldn't look like a complete idiot by trying to refute a theology whose fundamentals I didn't even understand. >> >> Well, then your assumption that there is a list of reals of length >> >> omega_1 is curious indeed. >> > >> > If you are curious, why don't you try asking a question, instead of >> > making declarations? What, exactly, do you find curious? You removed the quote I was replying to, so I don't recall. >> > Well, it might help if you identified some conceptual errors on my >> > part, instead of sending me off to read books. Obviously, you haven't >> > indentified any major flaws in my logic, or you would be pointing them >> > out specifically, wouldn't you? >> >> Oh but I have -- just pointed out one conceptual error above re CH. > Not really. Yes, really. >> Another recently identified is your belief that if there is no upper >> bound on the length of the strings in a set, then the set must contain a >> string of infinite length. > > That was not my statement, but someone else's paraphrase of their > understanding of my position. Pay attention. I have repeatedly stated > that N is infinite IFF N contains infinite n. I think you just want ONLY IF there, not IFF -- presumably you don't think that {m} is infinite just because m is one of your infinite numbers. Given that, since a set N of natural numbers infinite iff there is no upper bound on the size of numbers in the set, your principle would now appear to amount to pretty much the same thing as the "paraphrase" of your position above, only in terms of strings rather than natural numbers. >> Well, look. There are a lot of people who have a lot of years of >> advanced study behind them who are claiming that you are making some >> mistakes, and who, moreover, are going to some lengths to try to explain >> them to you. > > Actually, not. Generally my claims are refuted by being > mischaracterized and made to look stupid. That is not a valid > mathematical refutation, or an honest communication. No one's doing to you wrong on purpose. You are using words that have familiar meanings in mathematics but saying things that, relative to those meanings, are easily proved false and hence *do* look stupid. You can't *fail* to look stupid if you are going to make patently, provably false claims using the basic vocabulary of a scientific field without even going so far as to learn what researchers in the field mean by that vocabulary. I mean, surely you agree that is stupid, don't you? >> Metacomment: What could *possibly* drive someone who probably *knows* >> he doesn't understand Gýdel's theorem, and probably *knows* he doesn't >> have any idea what Gýdel proved about CH, to make pronouncements like >> this on a public forum? > > Conviction? Subway bombers have conviction, too. Unlike the principles of their bankrupt ideology, however, mathematical statements are capable of proof and disproof, but learning how to do mathematics requires discipline, hard work, and a measure of humility. Sadly, you appear bent instead on a cheap, spectacular, public self-destruction. Chris Menzel |