An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 20 Sep 2006 11:34 Mike Kelly wrote: > Tony Orlow wrote: >> Mike Kelly wrote: >>> Tony Orlow wrote: >>>> Mike Kelly wrote: >>>>> Tony Orlow wrote: >>>>>> Mike Kelly wrote: >>>>>>> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>>>>>> Mike Kelly wrote: >>>>>>>> >>>>>>>>> [ ... snip ... ] It's not clear to me that providing finite examples then >>>>>>>>> saying "obviously this holds for infinite cases too" without any >>>>>>>>> justification whatsoever should be at all convincing to anyone. >>>>>>>> It may be not clear to any mathematician, but it is clear to any >>>>>>>> scientist. The reason is that infinities do not really exist. >>>>>>>> They only exist as an attempt to make the "very large" rigorous >>>>>>>> in some sense. The moment you forget this, you get into trouble. >>>>>> Han, if I prove inductively, say, that 2^x>2*x for all x>2, do you find >>>>>> it objectionable to say that this also applies to any infinite value, if >>>>>> such a thing existed, given that any infinite value would be greater >>>>>> than any finite value, and therefore greater than 2? >>>>>> >>>>>>> But we are discussing whether there exists a uniform distribution over >>>>>>> the naturals. If you don't think this claim means anything at all then >>>>>>> why do you dispute it? If you reject the existence of the set of >>>>>>> natural numbers then you reject the set theory probability is based on. >>>>>>> So why bother to argue against individual theorems? You don't accept >>>>>>> *any* of probability theory. >>>>>> Just because someone disagrees with the transfinite portions of set >>>>>> theory doesn't mean they reject all of set theory. Clearly those of us >>>>>> who object do so on the basis of the conclusions drawn in infinite case, >>>>>> which derive from the axiom of infinity and/or the axiom of choice. >>>>> So, which do you reject? The axiom of infinity or the axiom of choice? >>>>> >>>>>> As >>>>>> far as probability goes, it certainly depends on the concept of sets, >>>>>> since probability more or less measures a subset of events with respect >>>>>> to the entire set of possible events. However, the same question remains >>>>>> as with the rest of transfinitology - is the cardinality generalization, >>>>>> based solely on raw bijection, really the most appropriate >>>>>> generalization from the finite to the infinite for sets? >>>>> Irrelevant. >>>> To what? >> Ahem! No answer? To a request for clarification of a curt dismissal? Hmmm... > > Irrelevant to whether there can be a uniform probability distribution > on the naturals. You know, the topic at hand? > > It's pretty rich for you to get uppity about one missed question on > what should have been an obvious point. You don't respond to salient > points /all the time/. Heck, in the last few days you've completely > ignored several of my posts then complained that I didn't respond to > your posts! While we're on the subject of unanswered questions : > > Which axioms or logic of ZFC set theory do you reject? The axiom of choice, and the axiom of infinity is insufficient. It is better to start with simpler axioms to build your theorems. > Do you understand now that the sum of an infinite series need not > exist? I understand that standard treatment does not assign a value to divergent series, but I also know it is possible, with a unit infinity, to represent many of them. > Do you admit that your attempt at a proof that the naturals can be > bijected with their powerset was totally bogus? No, but I admit that most of the naturals it maps to are not STANDARD finite naturals. > Do you understand that Cardinality doesn't claim to be the only or best > analogy to "size" for finite sets? Mathematicians claim that. If not, then why so much resistance to attempts to improve upon cardinality? It seems someone's toes are getting stepped on. When people use words like "same size", "equinumerous", etc, that's what they are implying, and it's only when backed into a corner that they admit there may be some difference. > >>>>>> Do we need to >>>>>> know the last element and exact range to derive a probability for >>>>>> something as simple as "n is a multiple of 3"? >>>>> No, but we need to know that it is possible to define a uniform >>>>> distribution on the set. >>>> Which requires an average, which requires a range. >>> What does "requires an average" mean? >> It means it requires a count and a sum, and the notion of division. > > *sigh*. So what does "requires a count and a sum, and the notion of > division" mean? And what does any of this have to do with probability? > Are we talking about probabiltiy theory here or your own vague ideas > about what you think probability theory should be? I thought we were talking about a uniform probability distribution. > >>> Loosely speaking, to define a uniform distribution to select an element >>> from a set one assigns a contsant probability to each element such that >>> they all sum to 1. No such contstant exists for countable sets. >> They cannot exist for "countably infinite" sets, since those have no >> upper bound (omega notwithstanding). Without an upper bound, there's no >> mean, and no distribution. >> >> Do they exist for "uncountably" (aka actually) infinite sets? > > Yes, continuous probability distributions. > >> Is there an average value of the reals in [0,1]? > > Yes, 1/2. > >> No, that also would require the conception of a value less than any finite, an >infinitesimal probability for each real, which would sum to 1. > > No, it requires that the integral is 1. A probability measure is > required to be 1. In the discrete case this means the summation of all > the elementary probabilities must be 1. For continuous probability > distributions, the integral must be 1. So, what is the chance that 1/2 is selected, for instance? > >> So, you probably reject that notion >> as well. > > Why? Because you cannot give a nonzero probability to any individual possibility, so the individual values cannot sum to 1. So, now you say there is a different approach that comes to the desired conclusion. So, you have two DIFFERENT probability theories. Why do you need two? Because you reject the notion of infinitesimals. > >> However, the average value of the reals in [0,1] is quite >> obviously 1/2. > > Sure. > >> So, you have a bit of a problem there. > >
From: Tony Orlow on 20 Sep 2006 11:43 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. > Oh? For what finite x is x-1=x?
From: Tony Orlow on 20 Sep 2006 11:58 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. > > - Randy > Given that all evens are natural, but only every other natural is even, is is logically deducible that the naturals are twice as numerous in any range as the evens. Intuition is a wholistic application of logic and association, not to be rejected unless proven contradictory. Tony
From: Tony Orlow on 20 Sep 2006 12:04 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. It is the set of all naturals, 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 is >> greater than all finite naturals, > > This isn't even meaningful using the ">" symbol > as defined on natural numbers. Sure it is. > > However it can be made meaningful if you define > ">" in terms of bijections. If you introduce a metric, sure. > > 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. > >> as any successor is greater then all >> those that precede it. > > It isn't a successor of any particular natural number. > So given that it is NOT a successor, how do you > think a property of "any successor" is relevant? Every ordinal is the set of all which came before it. Omega is AFTER the naturals in the quantitative order. > >> It is certainly a positive number, > > No it's not, as it does not fit the definition for any > positive number. > >> if it is a >> count or size of anything. > > A T-Axiom? It's basic logic. You don't have NEGATIVE sizes for your sets, do you? Not in standard theory. > >> If it is a positive natural greater then 1, > > It isn't a positive natural. > >> 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". > > It doesn't obey this general rule because it doesn't fit > the definitions of the things that do. That's not a > kludge. It's pointing out the "broccoli" does not > necessarily have the properties of things in the > set of mammals. 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. > > - Randy > Tony
From: Tony Orlow on 20 Sep 2006 12:12
imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> Han de Bruijn wrote: >>> Tony Orlow wrote: >>> >>>> Han de Bruijn wrote: >>>>> Precisely! Mathematicians get confused by the idea of a "bijection", >>>>> which is an Equivalence Relation, which in turn is a "generalization" >>>>> of "common equality" (yes: the one in a = b). But the funny thing is >>>>> that EQUALITY HAS NEVER BEEN DEFINED. > > Idiot. Do you know the definition of an equivalence relation? Do you > claim that bijection is not an equivalence relation? > > (I think that was a different crank - here's Tony...) > >> Consider the equally spaced staircase from (0,0) to (1,1), as the number >> of steps increases from 1 without bound. Is it the same as the diagonal >> line? > > What _exactly_ is "it" here? Idiot. What was the object referred to in the previous sentence? Do you not know how to correlate pronouns to their reference? > If I consider the set of staircases as the > number of steps increases without bound I get an unending set of > staircases. The only obvious singular object is the set, which is not > anything like a diagonal line (of course I don't think you mean this); > otherwise there are lots of staircases. Well, now you foam at the mouth > a bit... I mentioned ONE staircase, in the limit as the number of steps approaches oo. Don't play dumb. > >> Inductively we can prove that the length of the staircase is 2 at >> every step. Does it really suddenly become sqrt(2) in the infinite case? >> By the measures of point set topology, all points in the staircase >> become indistinguishable in location from the those of the diagonal, so >> by this thinking, all difference has disappeared, and the two objects >> are equal. However, using a segment-sequence topology, staircase n is >> the concatenation of n pairs of segments, denoted by x and y offset, >> described by {0,1/n} {1/n,0}, whereas the corresponding segments of the >> diagonal, between the points on the diagonal where perpendicular lines >> pass through the vertices of the staircase, are of the form >> {sqrt(2)/2n,sqrt(2)/2n}. The fact that the directions of the two curves >> are different at every point explains the difference in length, but this >> distinction cannot be detected by looking at pointwise location alone. > > Blabbley-blobbley. Wipe your mouth before you talk. I was reading the Wikipedia article on "Crank > (person)" today. Particularly the bit about cranks' incredible > over-rating of their own abilities. You seriously think you are so much > cleverer than the staff of every maths department in the world that you > alone can notice that every one of these staircases has length 2; you > think mathematicians in general are _that_ stupid? I think I have a nonstandard perspective which is at least equally valid as the standard transfinitology. > > Mike Kelly [I think] went all through the stuff about limits (that's > the mathematical term, with the normal mathematical meaning, not any of > your waffle) with you - to no effect, obviously, since overawed by the > power of your own brain, you don't have time to understand what the > term means. You seem to have some notion of your own (ice pose a > "Tlimit"), which is some property of a sequence such that it embodies > any property of all members of the sequence. Trouble is, in the end > this is just the sequence itself. You can't give two _different_ > sequences that have the same Tlimit. So a Tlimit captures no > generalisation at all. (You won't know this, but in a sense mathematics > is precisely the business of capturing generalisations.) I did give another curve with the same "Tlimit" as the staircase in the limit, which produced an interesting result, giving weight to the notion that an infinitesimal is something distinct from 0, whose square is not distinct from 0. I had it posted on my Cornell web site, but since I've left Cornell, I have to get a new web site together. I'll post a link when I get to that. In the meantime, your claim that my position was refuted is vacuous at best. > > Brian Chandler > http://imaginatorium.org > Tony |