Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 21 Jul 2005 00:13 In article <MPG.1d4866081dc65c12989f53(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > But the Peano axioms say that *all* nonempty sets of naturals have > > a smallest element. So if you say that there is no smallest infinite > > natural, then that implies that there are *no* infinite naturals. > > > > -- > > Daryl McCullough > > Ithaca, NY > > > > > all nonempty sets of finite naturals have a smallest element. All non-empty > sets of infinite naturals have a largest element. But as there are are no non-empty sets of infinite naturals.
From: Virgil on 21 Jul 2005 00:19 In article <MPG.1d489f7d7d5b57b989f63(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > 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 Right!
From: Virgil on 21 Jul 2005 00:21 In article <MPG.1d489f1a196cd916989f62(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > 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? Barb does not have to refute once more what so many others have already totally refuted.
From: Virgil on 21 Jul 2005 00:27 In article <MPG.1d487e8eef8126b1989f58(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d4816be96c6f78d989f36(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > Chris Menzel said: > > > > > > Pretty clearly, you aren't terribly well-educated in set theory. Don't > > > > you think you should understand a field before you try to point out its > > > > flaws? And don't you see that it makes you look rather silly? Would > > > > you consider attacking superstring theory without understanding basic > > > > quantum mechanics? > > > > > I have not immersed myself deeply in set theory, no. But, I can see > > > clearly > > > that some of the conclusions are wrong > > > > Without knowing how and why those conclusions have been reached? > > > > What sort of crystal ball are you using? > > > An infinitely average square black one made of liquid stone, not dissimilar > in > appearance to your head. TO has no concept of mathematics. TO swallows camels and strains at gnats. He has no sense of just how wrong he is about everything. One wonders how he manages to survive if his understanding of other fields is equally inept.
From: Virgil on 21 Jul 2005 00:28
In article <MPG.1d487edc1b8d1b9d989f59(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > stephen(a)nomail.com said: > > In sci.math Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > stephen(a)nomail.com wrote: > > > > >> It seems like a lot of the "anti-Cantorians" and other > > >> mathematical nay-sayers tend to start with their conclusions, > > >> and then try to work backward. The idea of starting > > >> with a fixed set of well defined axioms and working > > >> forward seems totally alien to them. > > > > > Nah, nah. First comes the theorem. And then comes the proof. > > > > > Han de Bruijn > > > > You cannot prove something without axioms. So if you > > start with your conclusions, you have to create some > > axioms before you can find a proof. Unfortunately they > > never seem to actually define any axioms. Look at Tony Orlow > > in this thread for a fine example of this sort of behavior. > > > > Stephen > > > I am working on a set of axioms. One of the central ones is N=S^L, which I > have > used repeatedly. It's difficult work, but it will be done over vacation in a > few weeks, hopefully. We look forward to a good laugh when they are published. |