Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 20 Jul 2005 18:48 imaginatorium(a)despammed.com said: > > > Tony Orlow (aeo6) 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. Not infinite. With a known end, or bound. I am > > sure you know what I mean. > > Hmm. Been here before... > > Does "with a known end" mean that you can name the known end of the set > of pofnats, which you tell us is finite? And tell us what its successor > is? And explain how exactly there is no contradiction with the Peano > axioms? The contradiction come from your decalration that they are all finite. I have already repeatedly pointed out the flaw in that proof. Go ask your infinite- series friends. They'll tell you. > > (For anyone who hasn't been following this: pofnats are the (plain old > finite) normal mathematical naturals; the Tonats are the Orlovian > naturals, which look a bit like the n-adics.) sort-of. Maybe a little more like the Phillian numbers ;) > > I don't suppose Tony that in your recess you've considered how you're > going to learn abstract algebra, and recreate it? Since you don't know > any set theory yet, it's going to be a big job... I started work on my paper. There's a lot to do. > > Brian Chandler > http://imaginatorium.org > > > > > > > > > > -- > > > Daryl McCullough > > > Ithaca, NY > > > > > > > > By the way, I don't suppose you would be going to GrassRoots this weekend? I guess that's a no. Daryl and I are in the same town. I am now leaving work for a long weekend of music, dance, art, peace and love, hopefully without any beliigerent young punks fighting over girls and beer. Have a nice weekend. Enjoy the moon. http://grassrootsfest.org/ > > -- > > Smiles, > > > > Tony > > -- Smiles, Tony
From: Tony Orlow on 20 Jul 2005 18:49 Virgil said: > In article <MPG.1d4830d09199666c989f3e(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > stephen(a)nomail.com said: > > > In sci.math Barb Knox <see(a)sig.below> wrote: > > > > In article <MPG.1d4726e11766660c989f2f(a)newsstand.cit.cornell.edu>, > > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > [snip] > > > > > > >>Infinite whole numbers are required for an infinite set of whole numbers. > > > > > > > Good grief -- shake the anti-Cantorian tree a little and out drops a > > > > Phillite. Here's a clue: ALL whole numbers are finite. Here's a > > > > (2nd-order) proof outline, using mathematical induction (which I > > > > assume/hope you accept): > > > > 0 is finite. > > > > If k is finite then k+1 is finite. > > > > Therefore all natural numbers are finite. > > > > > > Talking to Tony is a waste of time. He does not understand > > > induction and is a firm believer in "after infinity". He > > > is a fine example of the non-mathematical sort who complains > > > about Cantor. > > > > > > Stephen > > > > > Nice ad hominem. You never understood any of my points. Take your fingers out > > of your ears and stop with the "blah blah" > > Why should one even bother to listen to one who continually makes no > sense, and continually refuses to listen to what does make sense.? > > > TO's "points" have all been refuted. For example, Barb Knox, above, > shows succinctly why TO's "infinite naturals" are a delusion. > Barb Know never repsonded to my refutation of the simple proof she put forth, thinking I had never seen it. What about it Barb? Looking up infinite series? -- Smiles, Tony
From: Tony Orlow on 20 Jul 2005 18:51 Virgil said: > In article <MPG.1d4832e1f03a8d50989f3f(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Virgil said: > > > In article <MPG.1d4712ec2f75d957989f26(a)newsstand.cit.cornell.edu>, > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > Is this the > > > > same as the proof concerning the "uncountability" of the reals? > > > > > > Quite similar, but not identical. The basis is the same: showing that > > > any mapping from the smaller to the larger _must_ fail to be surjective. > > > > > And why is a lrger set necessarily uncountable, or unenumerable? > > The DEFINITION of a set being "non-denumerable" or "uncountable" is that > there be no surjection from N to that set. > > So that whenever one can prove that there is no such surjection to a > set, that set is, BY DEFINITION "non-denumerable" and "uncountable". > > > > > If you can > > draw bijections between naturals and evens and declare them the same, when > > one > > is obviously twice the size of the other, then why can't a similar bijection > > be > > created. After all, wouldn't you say that the set of all integral powers of 2 > > is a countable set? Or all log2's of natural numbers? In my book, there are > > 2^N > > log2's of natural numbers. That doesn't make it uncountable. It just makes it > > a > > bigger set. > > "Bigger" in the sense of no surjection from the "smaller set to the > "larger", is one thing, "bigger" in the sense of having the "smaller" > set as a proper subset is different. While these two measures happen to > coincide for finite sets, they do not coincide for infinite sets, as the > definition of infinite for sets should hint to you. > gee, they coincide for finite sets AND infinite sets, under Bigulosity, but I don't suppose you consider that extra consistency any sort of progress. -- Smiles, Tony
From: Robert Low on 20 Jul 2005 19:14 Daryl McCullough wrote: > I just meant that you can formalize and prove the claim > "Every nonempty set of naturals has a smallest element" > in 2nd order PA. OK, now *that* I can believe.
From: Daryl McCullough on 20 Jul 2005 19:08
> 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. >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). >When you actually "run" the proof, You don't "run" proofs. >#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. -- Daryl McCullough Ithaca, NY |