Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 25 Jul 2005 17:14 David Kastrup wrote: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > This is precisely the kind of question which cannot be > > answered. There is no bound to the lengths of the strings, but you > > claim they are finite. > > Can you tell us the difference between "arbitrarily large" and > "infinite"? Leading the discussion into a circle won't help. karl m
From: Chris Menzel on 25 Jul 2005 17:16 On 25 Jul 2005 12:40:12 -0700, georgie <geo_cant(a)yahoo.com> said: > So what you are really saying is that averything is an axiom. > > "whfh a3r 23r237818n df er rdq2" > > is an axiom in some formal system that (probably) nobody had > ever considered yet and never will beyond this thread. > > I would hardly call that rigorous, since i just pounded my fingers > on my keyboard to develop it. Neither would I.
From: Virgil on 25 Jul 2005 17:43 In article <MPG.1d4ef2d3df7d5616989f71(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > Is there a 1-1 correspondence between digital strings and whole numbers? Depends on what one means strings and by whole numbers. TO includes among his "strings" things which have infinitely many characters including a first and last, yet are supposed to be serially ordered with, except for the first, each having both an immediate predecessor, and, except for the last, an immediate successor. > > Can you have an infinite set of strings without infinite lengths? Everyone but TO can. But TO handicaps himself. > > How can you have an infinite set of digital whole numbers, without having > infinitely long strings representing infinite values? By allowing those strings to get arbitrarily long (no limit on length).
From: stephen on 25 Jul 2005 17:48 In sci.math Virgil <ITSnetNOTcom#virgil(a)comcast.com> wrote: > In article <MPG.1d4eb8a520ca63b1989f6a(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> 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? > We already have cardinalities, which are sufficient to our needs. > Let's see TO's definition of an "infinite" number. Haven't you been paying attention? According to Tony, an number is infinite if it is larger than every finite number, and a number is finite if it is smaller than every infinite number. Just another example of the sublime beauty of Orlovian mathematics. Apparently there are only a finite number of numbers that are smaller than every infinite number. Lets call this number X. X is smaller than every infinite number. X+1 is not a finite number, because there are only X finite numbers. Therefore X+1 is not smaller than every infinite number. In fact, given that X+1 is larger than X, it is larger than every finite number and is therefore infinite. On the otherhand, Tony insists there is no finite X such that X+1 is infinite, despite all his "proofs" relying on that "fact". Stephen
From: Virgil on 25 Jul 2005 17:54
In article <MPG.1d4efa2a5544d27d989f73(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > stephen(a)nomail.com said: > > In sci.math Virgil <ITSnetNOTcom#virgil(a)comcast.com> wrote: > > > In article <MPG.1d48308522352190989f3d(a)newsstand.cit.cornell.edu>, > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > >> Inductive proof proves properties true for the entire set of naturals, > > >> right? > > > > > Wrong! It proves things only for the MEMBERS of that set, not the set > > > itself! > > > > > Definitions (Cantor): > > > (1) a set is finite if and only if there do not exist any > > > injective mappings from the set to any proper subset > > > (2) a set is infinite if and only if there exists any > > > injection from the set to any proper subset. > > > Clearly then, a set is finite if and only if it is not infinite. > > > Definitions (Auxiliary): > > > (3) a natural number, n, is finite if and only if the set > > > of naturals up to it, {m in N: m <= n}, is finite > > > (4) a natural number, n, is infinite if and only if the set > > > of naturals up to it, {m in N: m <= n}, is infinite > > > > > If these definitions are valid, then it is easy to prove buy induction > > > that there are no such things as infinite naturals: > > > > > (a) The first natural is finite, since there is clearly no > > > injection from a one member set the empty set. > > > > > (b) If any n in N is finite then n+1 is also finite. > > > This is also while quite clear, though a comprehensive proof > > > would involvev a lot of details. > > > > I can flesh it out a bit. > > > > Inductive step. Show that if n is finite, then n+1 is finite. > > Proof by contradiction. > > > > Suppose that n is finite, but that n+1 is infinite. This > > means there exists a bijection f from { 1, 2, 3, ... n+1} > > to some proper subset S of { 1, 2, 3, ... n+1}. Without > > loss of generality we can assume that S does not contain n+1. > > > > If we apply the function f to { 1, 2, 3, ... n} we > > get the set S-f(n+1). Because S does not contain n+1, > > S-f(n+1) is a proper subset of {1, 2, 3, ... n}. This > > means there exists a bijection from {1, 2, 3, .. n} > > to a proper subset of {1, 2, 3, ... n}, which means > > that n is infinite which contadicts the assumption that > > n was finite. > > > > Stephen > > > Yes, it relies, as usual, on the lack of a largest finite integer, but that > doesn't mean that there are no infinite integers. But if 1 is a finite integer, and n+1 is a finite integer whenever n is a finite integer, by induction ALL natural numbers must be finite integers. Which means that thare are no infinite integers that are naturals. So wherever TO is getting his allegedly infinite integers, it is not from N. > There is no smallest infinite integer, but that doesn't mean there > are no finite integers either. Induction proves TO wrong, AGAIN! |