Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: imaginatorium on 20 Jul 2005 12:10 Daryl McCullough wrote: > 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? > > What definition of finite are you using? Tony has no clue what mathematics is, nor how it is done, so he doesn't normally bother with definitions. The closest we got from him for a definition of "finite" was that a finite number is less than an infinite one. And you can guess the "definition" of infinite. As best I can grasp it, the central principle of Orlovian pseudo-maths is that "infinite numbers", never clearly being defined, but reached by continuing from finite numbers through a "twilight zone" (whose existence is anecdotally stated), have essentially the same sort of properties as "ordinary numbers". The Orlow-refutation of the diagonal proof rests on the "fact" that a list of infinite sequences of digits is not a quarter-plane, as one might imagine, but a rectangle, of width P and height Q (P and Q being some Orlovian infinite numbers), so of course the diagonal hits the (infinite) side at some point. The mathematical concept of a sequence being endless means (to us) that there is no end, but that doesn't stop Tony using the end to prove something. You'll notice he gets a bit irritable when people point out that one of his "proofs" doesn't work because there *isn't* a largest integer (or whatever). Tony has any number* of "proofs" that an infinite set of natural numbers must include "infinite naturals", but these are generally circular. The one from "information theory" says that since there can only be a finite number of strings of finite length (even if the length has no limit), then to get an infinite set of numbers, you must include some that are infinitely long. The bit after "since" is a restatement of what he purports to prove, but he ignores people pointing this out. * He claims three, but since anything follows from False, surely such "proofs" can be generated without limit. Brian Chandler http://imaginatorium.org
From: Tony Orlow on 20 Jul 2005 12:18 David Kastrup said: > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > David Kastrup wrote: > > > >> Oh good grief. Successor in interest to JSH, are we? > > > > JSH is not an anti-Cantorian. So this argument, again, doesn't make > > sense and may only be useful for the purpose of insulting. > > JSH was the one who repeatedly threatened a day of reckoning when his > opponents would be cast from the ranks of mathematicians. > > And that is exactly what Tony was doing here. I quote what you > snipped: > > >>> I will have my web pages published before too long, so I am not > >>> getting into a mosh pit with you again right now. Just be aware that > >>> anti-Cantorians are sick of being called crackpots, and the day will > >>> soon come when the crankiest Cantorians will eat their words, and > >>> this rot will be extricated from mathematics. > > I never claimed you would be cast from the ranks of mathematics, but that you will see the errors that you are currently ignoring, and that the rot of Cantorian cardinality will be removed from mainstream thought and replaced with ideas that don't lead to absurdity like Banach-Tarski. I do see the ramifications of this nonsense in many areas. Until you can demonstrate that the theory is really correct, I am well within my rights to disagree with your axioms and conclusions, and if that right is challenged, I will continue to defend it and challenge your theory. It takes two to tango, and if you end up with people vowing vengeance, well hell, you probably deserve it. Then again, maybe JSH is mentally unstable, but then so was Cantor, and so was Godel. -- Smiles, Tony
From: David Kastrup on 20 Jul 2005 12:23 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > David Kastrup said: >> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: >> >> > David Kastrup wrote: >> > >> >> Oh good grief. Successor in interest to JSH, are we? >> > >> > JSH is not an anti-Cantorian. So this argument, again, doesn't make >> > sense and may only be useful for the purpose of insulting. >> >> JSH was the one who repeatedly threatened a day of reckoning when his >> opponents would be cast from the ranks of mathematicians. >> >> And that is exactly what Tony was doing here. I quote what you >> snipped: >> >> >>> I will have my web pages published before too long, so I am not >> >>> getting into a mosh pit with you again right now. Just be aware >> >>> that anti-Cantorians are sick of being called crackpots, and >> >>> the day will soon come when the crankiest Cantorians will eat >> >>> their words, and this rot will be extricated from mathematics. >> >> > I never claimed you would be cast from the ranks of mathematics, but > that you will see the errors that you are currently ignoring, and > that the rot of Cantorian cardinality will be removed from > mainstream thought and replaced with ideas that don't lead to > absurdity like Banach-Tarski. I do see the ramifications of this > nonsense in many areas. Until you can demonstrate that the theory is > really correct, I am well within my rights to disagree with your > axioms and conclusions, and if that right is challenged, I will > continue to defend it and challenge your theory. It takes two to > tango, and if you end up with people vowing vengeance, well hell, > you probably deserve it. Then again, maybe JSH is mentally unstable, > but then so was Cantor, and so was Godel. Trying to place yourself in good company? -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Daryl McCullough on 20 Jul 2005 12:18 Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> If TO's assumprtions were actually the case, there would have to be a >> finite natural so large that adding 1 to it would produce an infinite >> natural. But TO cannot produce either a largest finite nor a smallest >> infinite, so the set of all finite naturals is already big enough. >> >We have been through all this before. You lay these requirement on me, but when >I say you cannot have a smallest infinite omega 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
From: Daryl McCullough on 20 Jul 2005 12:34
Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >You ask this as if any set that is larger than any other set is >"uncountable". Do you consider the two terms, "larger" and >"uncountable" to be synonymous? Not quite. "Uncountable set" means "set with a larger cardinality than the set of naturals". Look, Tony. Your objections to standard mathematics all seem to revolve around disagreements about the meanings of words. Words such as "infinite", "finite", "larger", etc. *Real* mathematics doesn't depend on word choice. Can you express what you are trying to say without using any of those controversial words? Normal mathematics can. The use of the word "larger" to mean "having a greater cardinality" is just terminology. All of mathematics would go through just as well without *ever* using the word "larger". You could just as well use the word "more bloppity": By definition, a set S is said to be more bloppity than a set R if there is a 1-1 function from R to S, but there is no 1-1 function from S to R. 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. In contrast, your arguments are about nothing *but* terminology. To me, that shows that there is no actual content to your arguments. An actual mathematical argument does not depend on word choice. As a challenge, see if you can express your claims about infinite sets, or infinite naturals, or set size, or whatever, *without* using the words "infinite", "larger", "size", etc. -- Daryl McCullough Ithaca, NY |