Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 25 Jul 2005 14:19 In article <MPG.1d4eafed434d52d5989f64(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Sorry, I've been away..... Daryl McCullough said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > >How do we get an infinite set, then, if m<=n is finite for any > > >finite n in N? > > > > You get an infinite set by (1) Pick some starting number a. (2) > > Pick an operation f(x) that, given a number n, returns a new number > > that is greater than n. (3) Then form the set > > > > { a, f(a), f(f(a)), ... } > > > > That's guaranteed to be infinite. > Meaning it goes on forever? If it goes on forever, for an infinite > number of iterations, each time incrementing the value of the next > element (assuming your f() is successor/increment), then the value of > the next element will become infinite. If it becomes infinite by operating one step at a time, there must be a step at which it becomes infinite. Which step is that, TO? On the other hand, if "infinite" here only means unending, there is no problem, as the sequence merely does not ever end.
From: imaginatorium on 25 Jul 2005 14:25 aeo6 Tony Orlow wrote: > You don't know what an infinite number, set, string, tree, or process are? They > go on forever, without end, sometimes in more than one direction. Oh, that's odd. I thought you said that the reason the diagonal proof was "wrong", was that the diagonal hit the right hand side of the rectangle of infinite digit strings. But now you say that if a digit string is infinite it doesn't have an end. So how does the diagonal manage to stop at this end that isn't there? Brian Chandler http://imaginatorium.org
From: Tony Orlow on 25 Jul 2005 14:32 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. There is no smallest infinite integer, but that doesn't mean there are no finite integers either. -- Smiles, Tony
From: Tony Orlow on 25 Jul 2005 14:34 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. :) -- Smiles, Tony
From: Tony Orlow on 25 Jul 2005 14:40
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? I am not. > > >If b and n are known, then b^n is known. If b > >and n are finite, then b^n is finite. This is just using the > >standard definition of finite, independent of cardinality. > > There is no standard mathematical definition, other than the > one you've rejected. Your definition of finite vs. infinite sets is okay. I don't have an issue with it. It's your treatment of infinite sets which doesn't work for me. The bijection being used distinguishes the infinity. > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony |