Cantor Confusion
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From: Tony Orlow on 10 Dec 2006 14:29 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > >> In article <1165488738.492139.68600(a)73g2000cwn.googlegroups.com>, >> mueckenh(a)rz.fh-augsburg.de wrote: >> >>> Bob Kolker schrieb: >>> >>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>> One of many examples: The set {2,4,6,...,2n} has a cardinal number less >>>>> than some numbers in the set. This does not change when n grows (yes, >>>>> it can grow!) over all upper bounds. Therefore the assertion that the >>>>> set of all even natural numbers has a cardinal number gretaer than any >>>>> even number is false. >>>> The set of natural numbers has a cardinal greater then any set >>>> >>>> {1, 2, ... , n} for any integer n. >>> Wrong. >> It is WM who is wrong > > Wrong. >>> The set of all natural numbers contains only natural numbers. >>> These number count themselves by >>> |{1, 2, ... , n}| = n. >> For every {1,2,3,...,n}, there is an n+1 greater than n. > > Correct. The set is potentially infinite. There is no number aleph_0 or > omega counting all natural numbers. > ========================= > I agree. There is no smallest infinity, and no discernible end to R. >> Repeating a falsehood does not make it any less false. > > Why then did you repeat always that here are no balls in the vase at > time 0? Why do you repeat without reasonable arguments that there are > more paths than edges in the binary tree, although no path springs off > without its own egde? > ========================= > Again I agree. There are 9 n balls after n iterations, and there are half as many paths as edges or nodes in the tree, despite supposed bijection hat tricks. Banach-Tarski is ridiculous, and omega is a phantom. >> Why he maintains this this somehow bijects the individual branches with >> paths, is not clear. > > Because every branch consists of two edges, one for the each of the > resulting paths. > > ========================= > To put it another way, each right branch may be considered a continuation of the same original infinite path, like adding a 0 to the right of the decimal point - it doesn't change the value. The left branch from every node produces a new path, creating a new value with the concatenation of a 1 to the string. Therefore, for every two branches, there is one additional path. >> It is not the contents of a single line but the set of all lines that is >> being quantified, so that the finiteness of lines is irrelevant. > > Wrong. We know from every line that it is finite. therefore it is > uninteresting which line we consider. > > ========================== > > Is the diagonal longer than any line? Nope. >> Such faith in the illogical is charming, but doomed. >> ZF, or NBG, or something like them, will be around long after we are all >> gone. > > Of course, in the way as the old science astrology remains being around > here, never betrayed by the majority - but this majority is a quantity, > not a quality. > > =========================== > > Sure, it's a fun game, for the mystically minded. >> I did not claim to have personally shaken hands with each of them, as >> Ramanujan was rumored to have done, but I know where they live. > > You think you know where some of them live. And you know them as well > as you know the integer [pi*10^10^100] > > ============================= > >>> You cannot imagine the integer [pi*10^10^100]. > >> Why not, you just did! > > No. I did not. But I now understand why you think that you could know > natural numbers. You simply think "number" and imply that this covers > all numbers. You simply think "pi" and assert that you imagine the > number pi. You simply think "ZFC is free of contradictions" and so you > have proved that ZFC is free of contradictions. > > ================================== > Well, Wolfgang, I think it's valid, if you can formulate an expression in a mathematical language, to claim that's a number. The question is, what can be done with it? >> Sqrt(2) has a directly approachable decimal numerical address. > > Yes. But there are only countably many addresses. > > > Regards, WM > Perhaps, with a countable number of bits. But, there are more than any finite number of reals in the unit interval, and this is an infinite number. So, somewhere, there must be room for actual infinity, even if not within the confines of a countable language. Tony
From: Tony Orlow on 10 Dec 2006 14:32 Mike Kelly wrote: > Tony Orlow wrote: >> stephen(a)nomail.com wrote: >>> Han.deBruijn(a)dto.tudelft.nl wrote: >>>> stephen(a)nomail.com schreef: >>> Do you or do you not wish to abolish any mathematics >>> that involves infinity? If you are perfectly content >>> to let others freely explore whatever they wish, then >>> why are you so aggressive? >>> >> I believe Han agreed that, if he saw a more satisfying treatment of >> infinite sets, he'd be open to it. Of course, he's been rather resistant >> to my alternatives, but it seems everyone is, for one reason or another. > > Because they're impossible for anybody but you to understand and of no > apparent utility anyhow. > That's in the mind of the beholder - you. >> In any case, I think his objections are specific enough to warrant >> some attention. "Calculus XOR Probability" was about the loss of >> additive probability measure, when you have an infinite set of equally >> likely possibilities, as a result of the notion of aleph_0 elements, and >> its standard inverse, if there is such a thing, of 0% probability each. >> No sum of 0's can be anything but 0. Is it unreasonable to want to >> preserve additive measure within probability over an infinite set? I >> don't think so, and the answer to that issue was obviously to allow some >> infinitesimal probability for each natural. > > Doesn't work. Even in systems with infinitesimals, a countably additive > union of sets with measure zero has measure zero. There is no uniform > probability distribution over the natural numbers. Even if you "allow" > infinitesimals. > Measure zero is not the same thing as infinitesimal measure. Pay attention. >> In that case it can easily follow that the probability the n/3 e N is 1/3. > > Only by vigorous handwaving by people who really have no idea what they > are talking about. > Or by simple logic. >> Set theory "disagrees". Interpret that as you wish. > > Measure theory and probability theory do too, more to the point. > Great. Contradictions within mathematics. Wouldn't Hilbert be proud.
From: Virgil on 10 Dec 2006 14:33 In article <457c1249(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> Ugh, yes, I guess I have. The von Neumann ordinals appear to be the > >> vehicle connecting set membership and order this way. Okay. I don't like > >> it, but it works in its way. > > > > It is just supposed to work. No one is saying zero really is the empty > > set (whatever "really is" means). > > > > Then no one is saying the von Neumann successor ordinals "really are" > the naturals? Good. But everyone says they make an adequate model for the naturals, and for the ordinals. > > >> It seems like it would be better to have > >> another primitive, such as Peano's successor, than to use this strange > >> definition of the naturals, but I'll have to think about that. > > > > The idea is to be parsimonious. Since you can define the relation you > > want, there is no reason to be redundant by including another primitive. > > Doing so just makes things more complicated without any benefit. > > > > Not when the alternative is a construction that establishes the > equivalent of a new primitive through a dubious connection between > succession and containment. The only thing dubious about it is whether TO understood it. > What is more "parsimonious" in inventing > some weird model of the naturals and declaring that it exists, rather > than having succ() be a primitive relation? I don't see the advantage. As we already have sets, membership and inclusion, all the stuff for the vN system in every set theory, parsimony says don't add something else when what is already there is sufficient. > > If the sequence consists of segments of the form (0,x) or (x,0), there > is no segment which is diagonal in direction. If every segment in the > sequence is of the form (x,x), there is no segment which is not. This > information is not evident when the pairs describing the curve are > locations, because locations don't have direction. > > So, my question remains. If this is a valid formulation of the two > objects, and an explanation for Chas' counterexample to infinite-case > induction, where does this fit with set theory? I don't think the von > Neumann ordinals as a model of a sequence can suffice, since they only > allow finite values until a leap is made to the limit ordinals, and > continuity is violated. This is the problem I have with the vNO's, and a > large part of my problem with transfinitology. The notion that a > sequence must be "countable" simply is not correct in the bigger picture. > > >> So, the point set approach has that the same object has two different > >> measures, because it cannot distinguish between two objects which are > >> locationally the same, and directionally different. > > > > Then come up with an approach that does what you want. By "approach", I > > mean definitions (of objects, convergence, etc.) that let you prove the > > theorems you want. That's what everyone does. For example, Einstein came > > up with Brownian motion, but it wasn't clear how to mathematically model > > it. Weiner figured out how. > > > > Well, I'm suggesting a definition of the curve as a sequence of pairs > which denote xy offsets, rather than a set of pairs of xy coordinates. The standard definition of a curve is as the continuous image of a real interval. Such curves have 'directions' only at points at which the function describing them is differentiable, so in the limit, TO's step function would have no direction at any point.
From: Virgil on 10 Dec 2006 14:35 In article <1165759395.675493.173220(a)n67g2000cwd.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > > > > Ah, now don't act as if you didn't have those heated debates with some > > > manifest opponents of set theory, i.e. Wolfgang Mueckenheim. > > > > If you've been reading the threads, then you should know that WM has so > > far failed to present any mathematics at all. All he does is present > > incorrect arguments that he insists follow from the standard axioms and > > then proclaim: "Behold, standard mathematics is inconsistent." Big deal. > > Nonsense. The arguments presented by WM are quite reasonable. Not in mathematics. And that is the only relevant arena.
From: Virgil on 10 Dec 2006 14:39
In article <1165760388.618783.271900(a)n67g2000cwd.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > stephen(a)nomail.com schreef: > > [ ... Thank you, Stephen, for this quite detailed explanation ... ] > > But then: > > > This whole calculation, including time, can be modelled > > in set theory. > > It can't. Don't we have a debate with Wolfgang Mueckenhein about sets > that change in time? With a negative outcome? But a function whose domain is time can have different vales at different time without an unchanging functional relationship. > > Let us say that set theory is half the truth. Within set theory, any > function is a special relation between commodities and products, i.e. > domain and range. But the production process itself involves _labour_, > hence time, and this aspect is not covered by the static set theoretic > framework, where functions are reduced to "mappings" between sets. It is when one of these sets is a set of times. The illusion that time cannot be a variable in a functional relationship is an anti-mathematical idiocy. |