Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 20 Jul 2005 19:40 Alan Morgan wrote: > In article <MPG.1d485ba7151dad6989f4e(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > >Daryl McCullough said: > >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > >> > >> >>> There is nothing in Peano's axioms that states explicitly that all > >> >>> natural numbers are finite. > >> > >> Let's get more specific, and consider the sets S_n = the set of all > >> natural numbers less than n. Are you claiming that there is a natural > >> number n such that S_n is not finite? > >Actually, no, I am saying that for all finite n, S_n is finite, and that, if > >all n in N are finite, then so is N. Conversely, since S_n is finite for finite > >n, if S_n is infinite then n is infinite. > >> > >> What definition of finite are you using? > >Less than any infinite number. > > And the definition of infinite number is.... greater than any finite number? Having an arbitrarily large number of digits -- e.g. taking the output of Turing machines that don't stop as things. karl m
From: Chris Menzel on 20 Jul 2005 19:40 On Wed, 20 Jul 2005 20:55:58 +0100, Robert Low <mtx014(a)coventry.ac.uk> said: > Daryl McCullough wrote: >> That's not true. If S is an infinite set of strings, then there is a >> difference between (1) There is no finite bound on the lengths of >> strings in S. (2) There is a string in S that is infinite. > > Except that TO claims that (1) implies (2), though I can't > even get far enough into his head to see why he thinks it, > never mind finding his 'argument' convincing. This seems to be a fairly common element in crankitude. I've seen several folks argue here and elsewhere that there can be infinitely many natural numbers only if there is an infinite natural number. (Indeed, I think TO believes this, as I believe I saw reference to an "infinite natural" in one of his posts.) The origin of this idea sometimes seems to reside in imagination -- the poor afflicted fellows picture the number sequence as something like an endless string of beads that eventually disappears to nothing. There is thus no perceptual difference between *really really long* proper initial segments of the string and the entire string itself. So the entire string is the same sort of thing as its really really long proper initial segments. Other times, there seems to be some sort of a priori cardinality principle at work: for every set of natural numbers there is a natural number that numbers them. No finite natural number numbers all the finite natural numbers, so (obviously) there is an infinite natural number. Whatever. Kinda sad.
From: Jeffrey Ketland on 20 Jul 2005 20:20 Robert Low > Daryl McCullough wrote: >> In article <3k7o8jFt4lfgU2(a)individual.net>, Robert Low says... >>>Daryl McCullough wrote: >>>>But the Peano axioms say that *all* nonempty sets of naturals have >>>>a smallest element. >>>I'm not entirely sure what you're claiming here, but order isn't >>>even mentioned in the Peano axioms, so they certainly can't include >>>a statement that the naturals are well-ordered. >> I guess I was thinking in terms of second-order PA. > > It's pretty unclear in a lot of the thread when people > are talking about first order and when second order PA; > but I don't see how that helps. The only difference is > replacing the axiom scheme "If P is a one-place predicate, > then P(0) together with \forall n (P(n)->P(n+1)) entails > \forall n P(n)" with the axiom "If S \subset N, and > 0 \in S, and S is closed under successor, then S=N". > > We still don't have an explicit mention of order, so > the axioms can't just *say* than N is well-ordered, > even if they imply it. > > But maybe part of the problem is that the entire conversation > is taking place in an ill-defined context... True enough. Here are some details. One can define x<y in the first-order language of PA by the formula (Ez)(z =/=0 & y = x+z). The second-order statement "all non-empty subsets of N have a least element" can be "converted" to a scheme, the least number principle, namely, [i] (Ex)Px -> (Ey)(Py & (Az)(Pz -> z>= y)) which is then equivalent in (first-order) PA to the usual scheme. It's even more obviously equivalent to the scheme of total induction. Taking the contraposition, [i] is equivalent to [ii] (Ay)(Py -> (Ez)(Pz & z<y)) -> (Ax)~Px and, since P is schematic, this is equivalent to the scheme, [iii] (Ay)(~Py -> (Ez)(~Pz & z<y)) -> (Ax)Px which is equivalent to the scheme of total induction [iv] (Ay)((Az)(z<y -> Pz) -> Py) -> (Ax)Px This last is equivalent to the usual induction scheme, [v] [P0 & (Ax)(Px -> Psx)] -> (Ax)Px. There are four schemes of interest: - the usual induction scheme [v]; - the scheme of total induction [iv]; - the least number principle [i]; - induction "up to z". These are all equivalent (in first-order PA). More details in R. Kaye 1991, _Models of PA_, pp. 43-6. The least number principle does express that (N,<) is a well-ordering. --- Jeff
From: Daryl McCullough on 20 Jul 2005 20:01 Jeffrey Ketland says... > >Daryl McCullough > >> When I say "There are infinitely many natural numbers" I >> mean exactly "There exists a function f from naturals >> to a subset of the naturals". > >You must have mistyped, Daryl... >You mean the usual definition of Dedekind-infinity, >"X is Dedekind-infinite" >means >"There is an *injection* f from X to a *proper* subset of X" >(or, equivalently, "X is equinumerous with a proper subset of X"). Yes, I was typing too quickly. -- Daryl McCullough Ithaca, NY
From: Chris Menzel on 20 Jul 2005 20:19
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. > So, what does this have to do with assuming that a complete list of > the reals has aleph_1 members? Get it now? > The continuum hypothesis is mularkey as far as I can see. Yes, and we've seen how far you can see. > There is a whole spectrum of infinities between those alephs. 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. >> > 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? >> 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? Hope that's been explained now that I've explained a bit of the mathematics you didn't understand to you. >> > I have not immersed myself deeply in set theory, no. But, I can see >> > clearly that some of the conclusions are wrong, and in arguing >> > this, I have stumbled upon a few of the obvious flaws in the logic. >> >> David Kastrup has noted the problems with this reasoning. Many >> thinkers through the years have thought they clearly knew one thing >> or another, only to be shown wrong by deeper advances in the relevant >> discipline. As I noted earlier, actually learning the material you >> criticize as an intelligent but uneducated amateur will help you to >> see where your conceptual errors lie. > > 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. 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. Now, unfortunately, you are confusing your inability to see your errors when they are clearly pointed out to you with the idea that no one has found any. That is why I am, quite sincerely, pointing you to texts containing real mathematics, in the hope that some sincere study will enable you to see the errors you are making. >> > Hopefully I will find time soon to decipher the specific axioms of >> > ZF and see whether the roots of the problem lie there, or with >> > subsequent assumptions. >> >> Or with your own thinking -- an important part of being a scholar is >> always to bear in mind that one *might* be wrong. > > Yes, I have been corrected in the past, and that's okay. I haven't > been corrected correctly on this, and refuse to concede simply on the > basis of peer pressure. The flaws I see are glaring in my eyes, and > the refutations of them all have holes I can stick my hand through. 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. Moreover, it is more than evident, as you yourself admit, that you in fact are quite ignorant of the fields in which the things you criticize are studied; you don't understand some of their most elementary concepts. Now, just probabilistically, what are the odds that *all* of those folks (not to mention several generations of great mathematicians) are rather hopelessly confused about the foundations of their disciplines, and that you -- uneducated in those disciplines -- are right? Now, I grant you, it is not logically impossible -- perhaps God is playing a trick on everyone but you. But ponder the odds. >> > Now, if I came up with a theory that "proved" that 1=2, would you >> > believe it, at first glance? >> >> Of course not at first glance. But I would not dismiss it out of >> hand, either, given what we know from Gýdel about the unprovability >> of consistency. I would be greatly skeptical, of course, if the >> proof came from someone who obviously doesn't know much about >> arithmetic or set theory. > > Basically what Godel proved was that no system can prove itself > correct from within, so that proving any system correct becomes an > infinite regress into the systems from which it is derived or the > context in which it exists. Godel did some good work there. It's too > bad he got involved with the continuum hypothesis and ended up with > similar mental issues as Cantor. Ah, yes, working on CH causes mental illness. 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? Chris Menzel |