Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 26 Jul 2005 16:44 Daryl McCullough said: > Tony Orlow (aeo6) wrote: > > >The proof regarding strings is so simple, you really can't complain. N=S^L > >precisely describes the relationship between the number of strings N, the > >symbol set size S, and the length of strings L. > > But for the set we are talking about, there *is* no L. We're talking > about the set of *all* finite strings. That's an infinite union: If > A_n = the set of all strings of length n, then the set of all possible > finite strings is the set > > A = union of all A_n > = { s | for some natural number n, s is in A_n } > > This set has strings of all possible lengths. So there is no L > such that size(A) = S^L. If those lengths cannot be infinite, then the set cannot be either. Either you have an upper bound or you do not, and if there is no upper bound on the values of the members, then they may be infinite. If not, then what is the upper bound, and how do you have an infinite set of strings with only finite lengths? > > >I am not assuming anything except for this fact. > > You are assuming that every set of strings has a natural number L > such that every string has length L or less. That's false. I am saying that if L CANNOT be infinite, then S^L CANNOT be infinite, and the fact that so many find this impossible to understand demonstrates that Poincare was right, and Cantorian transfinite cardinality is a disease in mathematics. For finite S, S^L can ONLY be infinite with infinite L. Why is this so hard to understand? If S and L are both finite, then S^L is finite, isn't it? > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony
From: Virgil on 26 Jul 2005 16:42 In article <MPG.1d503dd82269292f989f93(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > In any case, sure, the program will spit out finite numbers, snce it is a > finite machine running in finite time. If the machine had infinite capacity > and > infinite funtime, it could conceivably produce infinite results. Not from the program described.
From: Tony Orlow on 26 Jul 2005 16:48 Chris Menzel said: > On Mon, 25 Jul 2005 11:27:28 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > > ...You cannot form an infinite number of strings with a finite > > alphabet, without strings of infinite length. > > Oh jeez, you're as clueless about formal languages as you are about set > theory, not that this is a surprise. Do you know that what you are > saying here is contradicted in the early chapters of any book on formal > languages? Really! Go have a look! Here, let me save you some > trouble: On page 1 of their famous standard text *Introduction to > Automata Theory, Languages and Computation*, Hopcroft and Ullman define > a "string" to be *finite* sequence of symbols. (Note: they don't think > the notion of string can't be generalized to the infinite, it's just > that in their text they are only interested in the finite ones.) Then, > on the very next page, they point out that "The set of palindromes > (strings that read the same forward and backward) over the alphabet > {0,1} is an infinite language." yes, I have looked around, and seen the damage done by this madness. When I was studying that, I shrugged at that nonsense. It's really not important when you are working with finite computers. It doesn't have any real-world implications, or at least not regularly. It's still incorrect, and it's still amazing that this sloppiness is tolerated. > > See? Do you realize what a fool you are making of yourself by making > assertions about set theory, transfinite arithmetic, formal languages, > computability, etc. that show you don't understand even the most > elementary concepts, or grasp the most elementary theorems, of set > theory, transfinite arithmetic, formal languages, computability, etc? You can call me a fool if it makes you feel better. You might want to read some Shakespeare and understand what the fool is. > > C'mon, stop embarrassing yourself. Go *learn* some mathematics before > you start spouting off about it. It's painful to watch. You are enjoying it, more than you should. > > Chris Menzel > > -- Smiles, Tony
From: Virgil on 26 Jul 2005 16:50 In article <MPG.1d5044b418d0c79d989f94(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: etc. > > > > In this representation each finite natural number is represented by a > > > > single finite string. Now how many of such finite strings are there, > > > > given that the stringlength is unbounded? > > > > > > This is precisely the kind of question which cannot be answered. > > > > Why can it no be answered? It is the same question as the question about > > the number of binary strings that have a leading 1 in a finite position. > Yeah, it's finite, but that's all that can be said. Actually it can be said, quite correctly, that there are more such strings that any finite natural. That TO cannot, or won't, say it is his error, not ours. > I guarantee you get the same result, if you do it right. If there is a choice between doing it right or doing it TO's way, do it right, even though that will often give you a result differing from TO's result. TO is still adding 1 to a finite number and getting an infinite number.
From: Tony Orlow on 26 Jul 2005 16:53
imaginatorium(a)despammed.com said: > Tony Orlow (aeo6) wrote: > > > I don't see where you pointed out any specific flaw, except to rant about your > > largest finite number again. > > No, well, I give up. Just for my curiosity, though, I still cannot > understand your point when you complain about "ranting about my[sic] > largest finite number". It has been pointed out to you so many times - > with absolutely no effect - that the Peano axioms (or any similar more > informal notion of pofnats) imply that there cannot be a largest > pofnat. Just tell me: do you claim... > (sigh) > (1) There _is_ a largest pofnat. no > (2) There is no largest pofnat (but the contradictions with your ideas > escape you) yes, please explain the contradiction, without the mantra. I have heard Virgil claim that I think there is one, or that I MUST produce one, if I am to claim there are infinite whole numbers. I see no such need. I ahve agreed that one cannot count finitely from the finite to the infinite, and it has been agreed that one cannot count down from the infinite to the finite. The first fact does not mean the infinite whole don't exist, any more than the second means that finite wholes cannot exist. So, where is the contradiction? > (3) The answer to "Is there a largest pofnat?" is somehow neither 'Yes' > nor 'No'. No, the answer is no, just like the answer to "is there a smallest infinite number?" There is no distinct line between the finite and infinite. That line is infinitely wide, and requires an infinite difference to cross. > > Thanks. > > Brian Chandler > http://imaginatorium.org > > -- Smiles, Tony |