An uncountable countable set
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From: Tony Orlow on 20 Sep 2006 10:15 Virgil wrote: > In article <45102b5d(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> I am well aware my position is "provably false", but I am >>>> also aware that that depends on the axioms assumed and the rules >>>> regarding logical inference. >>> No, your "position" is not coherent enough to be addressed as either >>> true or false. A bunch of gibberish doesn't admit of examination for >>> truth or falsehood. >> You are wrong. If I assert that S = proper_subset(T) -> |S|<|T|, then >> this is a basic axiom that one can accept which makes all of >> transfinitology untenable. > > Depends on how one defines |S| and |T|. > Given TO's assertion above, Card(S) and |S| are not the same, but that > in no way interferes with the validity of cardinality. If |S| is "the size of S, in count of elements", then it contradicts the use of cardinality as an analog for size in infinite sets.
From: Randy Poe on 20 Sep 2006 10:19 Tony Orlow wrote: > Randy Poe wrote: > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >> Mike Kelly wrote: > >> > >>> Infinite natural numbers. Tish and tosh. Good luck explaining that idea > >>> to schoolkids. > >> Look who is talking. Good luck explaining alpha_0 to schoolkids. > > > > I think I was 10 when I saw the proof that the rationals are > > countable, and first saw the notation "aleph_0". I don't remember > > having a problem with it. > > > > Perhaps you blocked it out. Perhaps you aren't telepathic. I can still remember the book, and the PAGE in the book, where I saw it. I don't believe my memory is faulty in the way you diagnose. > I know that, as soon as I was presented with > the "proof" that there are as many evens as all naturals, even though > that's only half of them, I detected a logical contradiction. Your intuition was bothered. That's not the same as "detecting a logical contradiction" as it involves intuition, not logic. The logic, i.e. series of deductions from starting axioms, is clear enough. - Randy
From: Tony Orlow on 20 Sep 2006 10:29 Mike Kelly wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> Mike Kelly schrieb: >> >>> mueckenh(a)rz.fh-augsburg.de wrote: >>>> Mike Kelly schrieb: >>>> >>>>>> Any set that can be established is a finite set. >>>>> Why? >>>> Look: If aleph_0 were a number larger than any natural number, then for >>>> any natural number n we had n < aleph_0. "For all" means: even in the >>>> limit. >>> OK so far. Every cardinal number which is a natural number is less than >>> aleph_0. >>> >>>> So lim [n-->oo] n/aleph_0 < 1 >>> Division is not defined for infinite cardinal numbers. >> Is that your only escape? If you dare to say that aleph_0 > n for any >> n e N, then we can conclude the above inequality. > > No, because division is not defined on infinite cardinal numbers. The > above inequality is meaningless. > >> But remedy is easy. >> Take lim [n-->oo] aleph_0 / n > 1 and reverse the following fractions >> analogously. >>>> If aleph_0 counted the numbers, for instance the even naturals, then we >>>> had for all of them >>>> >>>> lim [n-->oo] |{2,4,6,...,2n}| = aleph_0. >>>> >>>> This would yield lim [n-->oo] (2n/|{2,4,6,...,2n}|) < 1 >>>> >>>> But we have lim [n-->oo] (2n/|{2,4,6,...,2n}|) = 2 >>>> >>>> Therefore aleph_0 does not exist as a number which could be compared >>>> with other numbers. >>>>>> No. Just this is the point! The series 1 + 1/2 + 1/4 + ... is 2 (or at >>>>>> least as close to 2 as we like), not by definition and not by any >>>>>> axiom, but by rational thought. >>>>> Prove that to be the case without using any definition of what a series >>>>> is and without any axioms. >>>> Archimedes did so when exhausting the area of the parabola. In decimal >>>> notation 2 + 2 = 4, and in any system we have II and II = IIII. >>> In airthmetic modulo 3, 2+2 = 1. >> If you say "in arithemtic mod 3", then you imply that you subtract 3 >> from the true result as often as possible. It does not invalidate II + >> II = IIII, if you subsequently tale off III. > > Huh? The "true" result is that 2+2 = 1, if you are working in > arithmetic modulo 3. Or if it's 10 o'clock now and I wait 5 hours then > it is 3 o'clock. > > Your position seems very inconsistent. You claim that numbers have no > existence outside their representation. And now you are claiming there > exists a "true" arithmetic. > >>>> For self-evident truths you don't need axioms. Only if you want to >>>> establish uncertain things like "There exist a set which contains O and >>>> with a also {a}" then axioms may be required. >>>> >>>> Don't misunderstand me: I do not oppose the principle of induction but >>>> the phrase "there exists" which suggests the existence of the completed >>>> set. >>> Why do you object to this? >> Because of he proof above. > > It is not a proof. Division is not defined where either operand is an > infinite cardinal number. > If omega is the successor to the set of all finite naturals, it is greater than all finite naturals, as any successor is greater then all those that precede it. It is certainly a positive number, if it is a count or size of anything. If it is a positive natural greater then 1, then its reciprocal is a real in (0,1). To say that some count which is greater than any finite count does not obey this general rule is a kludge, like all the transfinite "arithmetic". Tony
From: Tony Orlow on 20 Sep 2006 10:37 Mike Kelly wrote: > Han de Bruijn wrote: >> Mike Kelly wrote: >> >>> Han de Bruijn wrote: >>> >>>> Mike Kelly wrote: >>>> >>>> >>>>> Han de Bruijn wrote: >>>>> >>>>> >>>>>> Mike Kelly wrote: >>>>>> >>>>>> >>>>>>> Han de Bruijn wrote: >>>>>>> >>>>>>> >>>>>>>> Mike Kelly wrote: >>>>>>>> >>>>>>>> >>>>>>>>> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>>> Mike Kelly wrote: >>>>>>>>>> >>>>>>>>>> >>>>>>>>>>> Infinite natural numbers. Tish and tosh. Good luck explaining that idea >>>>>>>>>>> to schoolkids. >>>>>>>>>> Look who is talking. Good luck explaining alpha_0 to schoolkids. >>>>>>>>> Sure, the theory of infinite cardinals is beyond (most)schoolkids. But >>>>>>>>> this is a bad analogy, because school kids don't need to know about >>>>>>>>> cardinals but they do need to know how to work with natural numbers. My >>>>>>>>> point, if you really missed it, was that Tony's ideas of "infinite >>>>>>>>> natural numbers" don't match up to our "naive" or "intuitive" idea of >>>>>>>>> what numbers should be - how we were taught to do arithmetic in school. >>>>>>>>> I for one don't understand what the hell an "infinite natural number" >>>>>>>>> is. And yet supposedly the advantage of his ideas are that they're more >>>>>>>>> intuitive than a standard formal treatment. >>>>>>>> My point is that the pot is telling the kettle that it's black (: de pot >>>>>>>> verwijt de ketel dat ie zwart is). Your aleph_0 is in no way better than >>>>>>>> Tony's "infinite natural number". >>>>>>> Your analogy is terrible, as usual. >>>>>>> >>>>>>> My point was that Tony's "infinite natural numbers" are not compliant >>>>>>> with everyday arithmetic. Aleph_0 is part of a formalisation that leads >>>>>>> to an arithmetic that works exactly as we expect it to. >>>>>> "... that works exactly as we expect it to". Ha, ha. Don't be silly! >>>>> So, what part of the arithmetic on natural numbers defined rigorously >>>>> as sets doesn't match up to the "naive" arithmetic we were taught at >>>>> school? >>>> I thought you meant the arithmetic with transfinite numbers. No? >>> >>> In what way is the arithmetic of transfinite numbers part of everyday >>> arithmetic??? >> Precisely! > > What the hell are you talking about? Arguing with someone who can't > speak English is getting aggravating. This isn't a language issue. Han is saying that transfinitology has NOTHING to do with everyday arithmetic. That's the point. It doesn't fir into mathematics. The conclusions are absurd. To quote George Boole, inventor of the system which allows you to confirm the deductive consistency of your axiom systems, in his "An Investigation Into The Laws Of Thought": "Let it be considered whether in any science, viewed either as a system of truths or as the foundation of a practical art, there can properly be any other test of the completeness and fundamental character of its laws, than the completeness of its system of derived truths, and the generality of the methods which it serves to establish." Where the conclusions are incorrect, where what is considered the "foundation" of mathematics contradicts many particular areas of mathematics, it can only be properly rejected as reflecting the fundamental truths upon which math is founded. > > I claim that Aleph_0 is part of a formalisation that leads to an > arithmetic on natural numbers that works just how naive arithmetic > works. Do you disagree? > Yes, wholeheartedly. In finite arithmetic, when you add a nonzero quantity, you increase the value - not so in transfinitology. You can remove elements, divide the set in half, double it, add elements, all without changing what is supposed to be the measure of the set. That's not how it works in the finite realm.
From: Han de Bruijn on 20 Sep 2006 10:40
Randy Poe wrote: > Tony Orlow wrote: > >>Randy Poe wrote: >> >>>Han.deBruijn(a)DTO.TUDelft.NL wrote: >>> >>>>Mike Kelly wrote: >>>> >>>>>Infinite natural numbers. Tish and tosh. Good luck explaining that idea >>>>>to schoolkids. >>>> >>>>Look who is talking. Good luck explaining alpha_0 to schoolkids. >>> >>>I think I was 10 when I saw the proof that the rationals are >>>countable, and first saw the notation "aleph_0". I don't remember >>>having a problem with it. >> >>Perhaps you blocked it out. > > Perhaps you aren't telepathic. > > I can still remember the book, and the PAGE in the > book, where I saw it. I don't believe my memory > is faulty in the way you diagnose. > >>I know that, as soon as I was presented with >>the "proof" that there are as many evens as all naturals, even though >>that's only half of them, I detected a logical contradiction. > > Your intuition was bothered. That's not the same as > "detecting a logical contradiction" as it involves > intuition, not logic. > > The logic, i.e. series of deductions from starting > axioms, is clear enough. Depends entirely on your conception of "clear enough". Read e.g. professor Mueckenheim's work on this: http://www.fh-augsburg.de/~mueckenh/ Han de Bruijn |