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From: Tony Orlow on 20 Sep 2006 15:23 Mike Kelly 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. It *is* the set of all finite naturals. By the way, omega is > an ordinal number, not a cardinal number. Yes, and the transfinite cardinals are based on the limit ordinals, eh? > >> 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?). Thank you. We're talking about the von Neumann ordinals because the alephs are the cardinals based on them, being the sizes of the limit ordinals, which are sets. That's approximately correct, no? So, these are supposed to be sets/sizes/numbers which have some relation to the finite naturals or reals, which really boils down to one of order. > >> as any successor is greater then all >> those that precede it. > > Sure... but omega still isn't the successor of anything. It's supposed to be the smallest infinite value, somehow less than which there is no other infinite value, because removing anything finite from it doesn't change it any. So, it's what comes right after (with some sort of a pause) all the finite ordinals. > >> 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?? It means it's not counting backwards. It's an absolute value, a distance, or difference, in units. (...and differences between differences...in units, of course) ;) > >> If it is a positive natural greater then 1, > > w isn't a natural. And I don't know what "positive" means for ordinals. There are only positive ordinals, in the von Neumann model. If you use the axiom of external infinity, that a set S is externally infinite, eoo(S), if for every yeS, ExEz x<y^y<z. Then you can define your '<' operator as decrement and '>' as inverse increment, and spew out the integers, with successor AND predecessor to every element. I don't see anything wrong with that. > >> then its reciprocal is a real in (0,1). > > What's the reciprocal of an ordinal? The reciprocal of any real on the line is a real on the line. Kepp it real, Man! > >> 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? > The distinction between cardinals and ordinals is artificial. Numbers are numbers. You either consider them in the infinite case, or you don't. I don't see how it's ultimately avoidable, given the continuum. But limit ordinals don't cut it, and transfinite cardinalities are for the birds.
From: MoeBlee on 20 Sep 2006 15:37 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> >> Actually infinite(S) <-> E seS A neN index(s)>n > >> >> Potentially infinite(S) <-> A seS index(s)eN ^ A neN E seS index(s)=n. > > > > So this is what you have now ('w' stands for 'N', which stands for?): > > Where do you see 'w'? > > > > > S is actually infinite <-> Es(seS & An(new -> index(s)>n)) > > I would rather not use 'w'. Sick to N. Why not? Okay, 'N' is fine. Except what do YOU mean by 'N'? > > S is potentially infinite <-> As(seS -> (index(s)ew & An(new -> Es(seS > > & index(s) = n)))) > > > > Okay, now I see that your 'actually infinite' is something like what > > set theory would describe as 'S has a member greater than any member of > > w'; and your 'potentially infinite' is what set theory would describe > > as 'S has only finite members but S has a denumerable number of finite > > members'. > > Okay. > > > Neither of those correspond to the usual senses of 'actually > > infinite' and 'potentially infinite'. > > In what way? Ask Wolfgang and Han whether they think so. Look up the terms 'actually infinite' and 'potentially infinite' in any reference work or overview and you'll see that your definitions do not correspond. > > Morevover, if you are going to > > allow the existence of a T-potentially infinite set (I'm going to use > > 'T' since your definitions don't correspond to usual senses) and just > > the basic set operations, then you are still going to have sets > > bijectable with proper subsets of themselves. And, you still will not > > have contradicted that an unbounded set must be infinite ('infinite' > > given the usual definition). > > No, but it will distinguish between potentially infinite sets like N and > actually infinite sets like R, which can have no algebraic relation. > That's because potentially infinite sets never are actually infinite, > but really finite but unbounded. Yes, with your definition, no T-potentially infinite set is T-actually infinite (given at least SOME ordinary sense of '>'). But what is "really finite"? You haven't defined 'finite' nor 'really finite'. > Well, I define N as the set of all sizes of sets, if you will, which do > not allow injections into themselves. Define 'sizes'. And I can only HOPE that 'injection' is meant by you in its ordinary set theoretical sense. Then PROVE that T-N exists. Oops, nevermind, you still don't have sufficient axioms set up to prove such things. > (and I understand you can't > define *everything* - just ask Virgil) Right. So you have to finally say what your primitives are. Then, since they are not defined, you have to give axioms regarding them that will make them work the way you want them to work to capture your intended meaning. MoeBlee
From: Tony Orlow on 20 Sep 2006 15:45 Mike Kelly wrote: > 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? I have nothing against set theory in general. It's the transfinite portions which are schlock. They lead to nothing useful, or even sensible. > >> 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? It means that it contradicts the greater part of our knowledge in the area, so there's something wrong, either with the assumptions/data, or the rules of inference. The first is a discovery problem, and the second a matter of mathematics and logic. I would suggest that, while there area couple of unresolved questions in logic, that the fault with transfinitology lies in its starting 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. What? No, you are missing MY point. It's not CONSISTENT with the rest of mathematics. It's a FICTION. > > I pointed out that your "infinite integers" contradict with everyday > arithmetic. Not all operations possible with finite numbers are possible with the T-riffic numbers, but I don't see as they contradict finite numbers in any way. > > Han responded with "what about aleph_0"? Yeah. Whattabout dat? ;) > > 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. > Yes. That's why the exchange above happened, which I'll paste here: Mike: 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. Han: "... that works exactly as we expect it to". Ha, ha. Don't be silly! Mike: So, what part of the arithmetic on natural numbers defined rigorously as sets doesn't match up to the "naive" arithmetic we wer
From: Tony Orlow on 20 Sep 2006 15:47 Mike Kelly wrote: > 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 } ? > That's a finite set. You don't need it.
From: Tony Orlow on 20 Sep 2006 15:54
MoeBlee wrote: > 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}. The size of the set x is the value, and it contains every natural less than x. Is this not the generalizable scheme? > >> 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. Aren't finite ordinals and cardinals and naturals the same thing? Perty much. > >> 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. I am well aware of the predecessor discontinuities which allow for "well" ordering. Remember Well Ordering the Reals? I think discontinuities in the continuum are in kind of bad taste. Really no need to pretend there's any smallest infinity, is there? > >> 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 > It doesn't seem to be "defined" anyway. :) ToeKnee |