Cantor Confusion
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From: David Marcus on 10 Dec 2006 14:39 stephen(a)nomail.com wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > stephen(a)nomail.com wrote: > >> But everything can be modelled as a set. You simply do not understand > >> what "foundation" means in this context. Any calculation can > >> be rewritten as a set theory problem. It would be long, cumbersome, > >> and impractical, but it could be done. Just as an computer program > >> can be transformed into a Turing machine. > > > Yes, but I think it is a bit simpler than that. The study of algorithms > > involves the study of certain functions on certain number systems. Such > > functions and numbers can be handled in ZFC. That's the real point of > > the construction of the natural numbers from sets: to show that set > > theory can be used as the foundation for arithmetic and hence analysis, > > etc. However, if you are doing numerical analysis or calculus, you think > > of N, Z, Q, R, C, etc. as primitive objects. You don't care that they > > can be modelled as sets. > > But writing down all the details using nothing but sets would > be cumbersome. Just saying that numbers can be represented as sets > is not the same as actually demonstrating how a calculation would > be represented using sets and nothing but sets. As I said, it > is like writing a Turing machine that computes a Fourier transform. > Anyone who understands the theory of computation knows that it > can be done, but nobody is likely to do it, especially just to > satisfy who has no understanding of the theory of computation. True, but all mathematics is built up from more primitive concepts. We can prove that something can be done without actually doing it. -- David Marcus
From: Tony Orlow on 10 Dec 2006 14:40 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > >> > > Pray reread what I wrote: "the nodes can be made to represent numbers in >> > > your tree". That is an easy exercise, I even did show it. The same for >> > > the edges, I did show that too. So actually the nodes and edges also >> > > represent numbers in some way. >> > >> > In the same way as the first few digits of a real number represents a >> > number. 3.1, 3.14, and so on represent numbers in some way. But that is >> > not at all important or interesting for the tree argument. >> >> So why did you state that I erronously believed that nodes represent >> numbers? > > Because you erroneously did. (The real numbers in the tree are all > represented by infinite paths like > 3.1000... > 3.14000... > etc.) > Correct, though I might restrict that to fractions. One needs to choose an interpretation of the tree and stick with it. One can view each infinite path as a real in [0,1), in which case each level of the tree represents a bit position, and each edge/node on that level represents a choice of 0 or 1 for that bit position. If the nodes represent values, you have a quite different situation. >> > > > Not at all! I represent numbers by standard binary notations. >> > > >> > > It is using the limits where you are doing something non-standard. >> > >> > I do nothing. The tree cares that even in the limit the number of paths >> > cannot become uncountable. 2^n remains the cardinal number of a >> > countale set, even in the limit n --> oo. That's why I devised the >> > tree! >> >> And it is exactly that what is wrong. For each finite n 2^n is the >> cardinal number of a countable set (even of a finite set), that does >> not make something like that also true in the limit. > > What limit are you talking about? My statement is true for each finite > level n. There are no others. > Agree. For all n in N it is true. There is no point within N where its power set is more than countable. For finite n, 2^n is finite and log2(n) is finite. >> It is easy >> enough to construct a bijection between the natural numbers and the >> edges, because the edges are countable. > > After all you have grasped that too? Until now you denied. > >> Contrary to what you write >> elsewhere, you have *not* constructed a surjection from the edges to >> the paths. If you think you did that present us with an edge that >> maps to 1/3, and show how the mapping is constructed. > > Please give me all the bits of 1/3. Then I will show the bijection. > > ============================= > >> It is easy enough to construct a surjection from the edges to the >> binary rational numbers with terminating expansion. But indeed, WM >> does not show a surjection at all. > > Indeed you are unable to understand fractions? > > Regards, WM >
From: Virgil on 10 Dec 2006 14:41 In article <1165760559.675770.203980(a)f1g2000cwa.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > > > Duh! Anyone could have told you that functions are a plausible > > foundation to build mathematics on. However, I'm not sure if this has > > been worked out in detail. In truth, most working mathematicians don't > > really care what foundation is used. > > If you do not really care, why are you so offensive then? It is HdB et al. who is on the offensive, and Marcus is merely defensive.
From: Mike Kelly on 10 Dec 2006 14:44 Tony Orlow wrote: > 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. And everyone else, as you admit. Either your ideas aren't worth a damn or you're incredibly poor at explaining your mathematical ideas. Well, you *are* incredibly poor at explaining your mathematical ideas; maybe you are onto something after all! > >> 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. The countable union of sets of singleton infinitesimals has zero measure. Pay attention. You really have no idea what you're talking about here. You simply don't know enough to make even vaguely sensible claims. > >> 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. Nope. You have no idea what you are talking about with regards to either probability theory, measure theory or infinitesimals. > >> 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. Uh? "Measure theory, probability theory and set theory all 'disagree' with a uniform distribution on the naturals. This implies a contradcition within mathematics." Baffling non sequitur. -- mike.
From: Virgil on 10 Dec 2006 14:46
In article <1165761763.908889.34550(a)80g2000cwy.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > stephen(a)nomail.com schreef: > > > You are the one who is trying to restrict peoples' freedoms. You > > cannot name a single piece of mathematics that is "forbidden" > > by set theory. > > Oh yes, I can: > > Let P(a) be the probability that an arbitrary natural is divisible by > a fixed natural a. Then P(a) = 1/a . Forbidden by set theory. HdB is again wrong. It is not forbidden by set theory itself but by the standard definition of probability within probability theory. > > http://groups.google.nl/group/sci.math/msg/b686cb8d04d44962 > > > You on the other hand want to abolish any mathematics > > that involves infinity, which is a severe restriction on others > > freedom of thinking. In short, you are a hypocrite. > > No. I am a truth seeker, against all odds. HdB has certainly stacked the odds against his ever finding truth, at least re mathematics. > > > I would imagine your understanding of Object Oriented Programming > > is as woefully garbled as you understanding of set theory. > > I've done quite some of it for a living. And I'm still alive and well. Not enough of it to have noticed that it is not a part of mathematics, apparently. > > Han de Bruijn |