Obections to Cantor's Theory (Wikipedia article)
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From: Barb Knox on 25 Jul 2005 22:10 In article <1122338688.718048.162860(a)g47g2000cwa.googlegroups.com>, malbrain(a)yahoo.com wrote: >Barb Knox wrote: >> In article <MPG.1d4ecd45545679a8989f6b(a)newsstand.cit.cornell.edu>, >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> [snip] >> >> >keep in mind that >> >inductive proof IS an infinite loop, so that incrementing in the loop >> >creates >> >infinite values, and the quality of finiteness is not maintained over those >> >infinite iterations of the loop. >> >> Using your computational view, consider the following infinite loop >> (using some unbounded-precision arithmetic system similar to >> java.math.BigInteger): >> >> for (i = 0; ; i++) { >> println(i); >> } >> >> Now, although this is an INFINITE loop, every value printed will be >> FINITE. Right? > >Not so fast. The behaviour of incrementing i after it reaches INT_MAX >is undefined. Not so fast yourself. You missed the part about "unbounded-precision arithmetic system", which has no max. -- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | -----------------------------
From: Virgil on 25 Jul 2005 22:15 In article <MPG.1d4f1bfe605de2db989f80(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Dik T. Winter said: > > In article <MPG.1d483583ff4dfb97989f41(a)newsstand.cit.cornell.edu> Tony > > Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > ... > > > > Now this case is quite dissimilar. Cantor's proof (about the > > > > cardinality > > > > of powersets vs. the cardinality of the base sets) is indeed simple > > > > and > > > > can be written in very few steps. > > ... > > > You know, if the only conclusion drawn from Cantor's proofs was that a > > > power > > > set is necessarily larger than its base set, I would have absolutely no > > > problem. > > > > Actually that is the only conclusion. But that conclusion can be shown to > > be equivalent to other conclusions. > > > > > That fact is always the case and I don't dispute it. Infinite sets > > > of > > > finite natural numbers, on the other hand, are self contradictory, > > > > Yes, you have told that already numerous times without actually showing > > that is true. But let us go from the binary to the dyadic notation. > > In that case the digits used are 1 and 2. 0 can not be represented, but > > each natural number can be represented in only *one* way by a string of > > those digits. A short table to show the idea: > > 1 = 1 > > 2 = 2 > > 3 = 11 > > 4 = 12 > > 5 = 21 > > 6 = 22 > > 7 = 111 > > 8 = 112 > > 9 = 121 > > 10 = 122 > > etc. > > In this representation each finite natural number is represented by > > a single finite string. Now how many of such finite strings are > > there, given that the stringlength is unbounded? > 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. What number am I supposed to use to calculate this set > size? It is the same "number" used to designate the number of members of N. As anyone with any common sense ought to be able to figure out. > Still, the question is interesting if infinite digits are allowed. Does TO mean that he is allowed to have one digit that is infinitely large or infinitely many finite digits in one number, or something else entirely? > > If you allow infinite strings and values, then I say there are N > numbers, by definition. And I say that if you allow infinite strings there are >= Card(P(N)). > > You are missing the point. Each set "larger" in some sense than > > the naturals as mathematicians think about them is (by definition) > > uncountable. > Sure, unless you consider sets like the halves or square roots to be > larger than N, which I do. But then TO every day believes 8 impossible things before breakfast, just to loosen up. > > > > Wrong, the diagonal proof proves the same thing as the proof that > > the powerset of a set is larger than the base set. The reals can > > be put in a 1-1 relation with the powerset of the natural numbers. > > (But only when you consider natural numbers in the sense of the > > mathematicians.) > > > And if you allow infinite naturals, then the naturals can also be put > in 1-1 correspondence with the power set of the naturals via binary > strings. If you allow infinite naturals, all sets can be shown to be finite, and simultaneously infinite, and anything else you want.
From: Virgil on 25 Jul 2005 22:17 In article <MPG.1d4f1c868f1d35f3989f81(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d485c6dc91e5c36989f4f(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > What it really shows is that digital systems with a given number of > > > digits have more strings than digits. it is not necessary to have > > > aleph_0 digits, if you allow for smaller infinities. > > > > Smaller than what? The set of naturals is as small as infinite sets get. > I disagree, obviously. TO gets off by being disagreeable. > > TO seems to be saying that because the set of all naturals has no limit > > on thet number of digits needed that there must be some one neatural > > with no limit on the number of digits needed, but it does not follow. > I never said that. I din't say you actually said it. Read my words. > > > > It is just another instance of TO's quantifier dyslexia.
From: Virgil on 25 Jul 2005 22:22 In article <MPG.1d4f1cac546f2c77989f82(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d485d5b56de5b6989f50(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > And yet, one cannot apply an increment an infinite number of time without > > > adding infinity. > > > > > > That TO says it cannot be done does not prove that it can't be done. > > > > Anyone but TO, and a few others of equally limited capabilities, can do > > it quite easily. > > > You can add an infinite number of 1's and get a finite result? Wow, you're > good! Did not say that. TO apparently cannot read very clearly. As I am not adding 'infinity' at any one step my unlimited but not infinite number of add-1 steps does not at any point add up to more than a finite number. Unless TO can tell me at just which step adding 1 to a finite number produces an infinite number. Can he?
From: Barb Knox on 25 Jul 2005 22:27
In article <ITSnetNOTcom#virgil-0F5C2E.14021925072005(a)comcast.dca.giganews.com>, Virgil <ITSnetNOTcom#virgil(a)COMCAST.com> wrote: [snip] >If the naturals satisfy the Peano properties, then any non-empty subset >of the naturals will have a smallest (first) member. > >If the set of infinite naturals is not empty, it must have a smallest >member. Subtract one and that must be the largest finite natural. But note that the 1st-order Peano axioms do allow non-standard models with "infinite naturals", such as these 2 lines together: 0 1 2 3 ... ... w-2 w-1 w w+1 w+2 w+3 ... Note that there is no smallest "infinite natural" here. This doesn't violate 1st-order Peano well-foundedness because the set of just the "infinite naturals" (those with a 'w') can not be defined in the first plase using 1st-order logic (on account of compactness). >Unless TO can produce evidence either of a largest finite natural or a >smallest infinite natural, he argues in violation of the Peano >properties. Not necessarily.... >So that whatever TO's "numbers" are, they are not the Peano naturals. Note that if TO actually took the trouble to learn some axiomatic mathematics he would likely find non-standard Peano models interesting. But of course as it is all he can do is handwave and bluster. -- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- |