An uncountable countable set
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From: Mike Kelly on 20 Sep 2006 13:32 Tony Orlow wrote: > 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 isn't. It *is* the set of all finite naturals. By the way, omega is an ordinal number, not a cardinal number. >it is greater than all finite naturals, Since we're apparently talking about ordinal numbers now (why?) I'll grant that omega is greater than all finite naturals under some ordering (inclusion?). > as any successor is greater then all > those that precede it. Sure... but omega still isn't the successor of anything. >It is certainly a positive number, if it is a > count or size of anything. What does it mean for an ordinal to be a positive number?? > If it is a positive natural greater then 1, w isn't a natural. And I don't know what "positive" means for ordinals. > then its reciprocal is a real in (0,1). What's the reciprocal of an ordinal? > 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". Counts? So we're back to talking about cardinals again? -- mike.
From: Mike Kelly on 20 Sep 2006 13:41 Tony Orlow 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 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. Even though set theory leads to an arithmetic on natural numbers that is identical to 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, What does it mean for a conclusion to be incorrect? That it is not a logical consequence of one's assumptions? >where what is considered the > "foundation" of mathematics contradicts many particular areas of > mathematics, For example? > it can only be properly rejected as reflecting the > fundamental truths upon which math is founded. (I think you meant to say "not reflecting") What fundamental truths would those be? > > 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. That's not how it works in "the finite realm" in set theory, either. Apparently you're completely missing my point, too. I pointed out that your "infinite integers" contradict with everyday arithmetic. Han responded with "what about aleph_0"? My reply is : aleph_0 doesn't contradict with everyday arithmetic. The arithmetic on natural numbers in set theory is identical to everyday airthmetic on natural numbers. Since everyday arithmetic doesn't include arithmetic on infinite cardinal numbers, it seems somewhat irrelevant to point out that you find arithmetic on infinite cardinal numbers unintuitive. This doesn't relate to everyday arithmetic in the slightest. -- mike.
From: Mike Kelly on 20 Sep 2006 13:45 Tony Orlow wrote: > Virgil wrote: > > In article <450d5f76(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> Mike, you haven't responded to my use of IFR > > > > An IFR, being dependent on order relations, at best measures order > > relations, not their underlying sets. > > Funny how it DOES measure the sizes of sets perfectly in all finite cases. So... how do we use IFR to tell us the size of the set {sqrt(17), Pi, e, {}, 42 } ? -- mike.
From: MoeBlee on 20 Sep 2006 13:46 Tony Orlow wrote: > > It isn't. What does "successor to N" even mean? > > Ask von Neumann. It is the set of all naturals, What are you talking about? The successor of w is not w. The successor of w is wu{w}. >and any ordinal being > the set of all preceding naturals, What are you talking about? An ordinal is the set of all preceding ordinals, not necessarily the set of all preceding naturals. > omega is the set of all preceding > naturals. It's larger than all naturals. DO you disagree with that > simple statement? No. But you must be careful not to take 'preceding' in the sense of 'immediately preceding'. Every member of w prededes w (as 'precedes' just has the sense of 'less than' in the ordering, which, in the case of ordinals is taken as the membership relation) but no member of w is an immediate predecesor of w in the sense of w being a successor to some member of w. > No, but if I say I have a science of all life, it should apply equally > to broccoli and mammals, and if I have a science of animals, it applies > to mammals, but NOT broccoli. So, if I have a rule for numbers, which > aleph_0 doesn't obey, I don't consider it a number. Fine. Don't consider it a number. Set theory does not depend on the word 'number'. MoeBlee
From: Tony Orlow on 20 Sep 2006 13:47
Randy Poe wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> 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 isn't. What does "successor to N" even mean? >> Ask von Neumann. > > I consider that a crackpot response. First the > crackpot misconstrues some paper they're reading > (by Cantor or Einstein, say), then when pressed > they respond "are you arguing with Cantor/Einstein?" > > I'm not asking von Neumann, I'm asking you. You > used a phrase. Did you have any idea what you > meant by that phrase? > >> It is the set of all naturals, > > N is the successor to N? Omega is successor to N, and the size of N, in the von Neumann system. > >> and any ordinal being >> the set of all preceding naturals, omega is the set of all preceding >> naturals. It's larger than all naturals. DO you disagree with that >> simple statement? > > It contains terms whose meaning we do not agree on. > > What is the test for "largeness"? Number of successions, in this case, resulting in the value. A larger number of successive increments produces a larger value. Can you really not say that an infinite count is greater than any finite count? That seems to be the basis for omega. > >>>> it is >>>> greater than all finite naturals, >>> This isn't even meaningful using the ">" symbol >>> as defined on natural numbers. >> Sure it is. > > We have a set N. We have an operation given > meaning on elements of the set N. It is not meaningful > on things not in set N. Even if you say "sure it is". > Asserting "sure it is" does not actually assign > a meaning to it. It gets a meaning only when you > actually go through the process of GIVING it > one. There are a number of ways to do that. "Larger" can mean "farther from 0", geometrically. "Larger" has no meaning without some kind of measure, but measure is really what any true number represents. > >>> However it can be made meaningful if you define >>> ">" in terms of bijections. >> If you introduce a metric, sure. > > Again you say "sure" too quickly. > > Is that an agreement that for ordinals, we will > use bijection to define ">"? Specifically, that > "A > B" means that any mapping from B to A > will leave elements of A unmapped? That's the way ordinals work. I'm not redefining ordinals. I might as well rewrite the Book of Mormon. I am saying that taking infinite case induction as a fundamental fact, the system of von Neumann ordinals as a model of the naturals can't follow. > >>> You are working with an undefined term, with undefined >>> operations, and then trying to draw conclusions >>> as if you'd defined those things. >> If an infinite number is not greater than a finite number, then it's not >> a number at all. > > Precisely. It doesn't fit the definitions for natural number, > real number, or complex number. It's a new kind > of thing, with new properties. Did you notice the "if"? You agree with the conclusion that your "transfinite cardinalities" are not numbers? That's a first. > > The operators like ">" which apply to natural numbers > don't apply here. You have to DEFINE them. They > don't automatically get meaning without new definitions, > because the old definitions |