Cantor Confusion
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From: MoeBlee on 30 Nov 2006 13:38 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > In particular because there is neither a stadard model nor a > > > > > non-standard model of ZFC. > > > > > > > > You haven't proven that there is no model of ZFC. > > > > > > Mathematics exists nowhere except in the minds of mathematicians. At > > > present there is no mind with such a model. So, where should it exists? > > > > As I said, you've not proven that there is no model of ZFC. > > I did not say that I had proven that. I didn't say that you had said that you had proven that ZFC has no model. Rather, you just claim it to be true that ZFC has no model, and it is MY remark that you have not proven it. > I said "In particular because > there is neither a standard model nor a non-standard model of ZFC." Are > you really incapable of understanding such simple sentences? Your posting "there is neither a standard model nor a non-standard model of ZFC" entails that you are claiming that there is no model of ZFC. And I have simply pointed out that you have not proven that claim. > > > > > And in particular because the expression "contradiction" is not pat of > > > > > ZFC. > > > > > > > > In Z set theory, we can formulate definitions of 'S is a contradiction' > > > > and 'T is inconsistent'. > > > > > > One could, but one wouldn't. Never! > > > > WRONG. We DO. > > > > You're an ignoramus. > > Do you really believe that set theory becomes more popular if its > adherents turn out to behave like undisciplined children without any > self-control like you or Mr Bader? I've never taken a job as the public relations manager for Set Theory Inc. My goal is not to make set theory more popular, but rather to speak my mind. What is puerile is your entire approach, based on maintaining your ignorance, to discussing set theory, which is the most salient thing that comes through your postings, and is thus mine to comment upon. When you claim things that are patently false such as that in set theory one would never formulate definitions of 'contradiction' and 'inconsistent', or to deny that in set theory we prove that the set of naturals is equinumerous with the set of rationals, then you can pretty much count on being called for being the ignoramus that you are. MoeBlee
From: MoeBlee on 30 Nov 2006 13:53 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1164816790.379338.139370(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > MoeBlee schrieb: > > > > > I think if one swells to explosion about his knowledge of set theory, > > > > > he should at least know the very foundation. But I know, that you do > > > > > not even understand the simple texts of Fraenkel et al. > > > > > > > > No, YOU radically MISunderstand what Fraenkel, Bar-Hillel, and Levy > > > > wrote. > > > > > > How can you judge about that without the slightest idea of what they > > > wrote? Only for the lurkers: Fraenkel et al. write: "Platonistic point > > > of view is to look at the universe of all sets not as a fixed entity > > > but as an entity capable of "growing", i.e., we are able to "produce" > > > bigger and bigger sets." So a set (like the set of all sets) is not a > > > fixed entity. > > > > There is nothing in that that shows that a set is not a fixed entity. > > The universe of all sets can grow. Define: "The universe of all sets is > called the set of all sets", and you see it. As to what Fraenkel, Bar-Hillel, and Levy wrote, they are underscoring the fact that different axioms yield different universes of sets. That is what they mean by the universe of sets "growing" (scare quotes in original text). > > You are able to produce bigger and bigger sets, but they are all > > different. > > That is a matter of definition. If you consider a fixed set then it is > fixed. Small wonder. If you consider a variable set then it is > variable and perhaps changes its cardinal number. You can go ahead and provide some theory in which there are such variable sets. Meanwhile, there is nothing like it in Z set theories. And Fraenkel, Bar-Hillel, and Levy's comments do not contradict this. > An easy example which > should not escape you: The set of states of the EC has been growing and > probably will continue to grow. Which you'll have a hard time proving to be a set in Z set theory. Of course no one denies that the everyday, non-mathematical, sense of 'set' includes the idea that many everyday conceived sets have changing membership. But that is distinct from the mathematics of Z set theory, which has an axiom that determines that the sense as applied to such theories is distinct from the everyday sense. No one is stopping anyone from formulating a mathematical theory in which sets have changing membership; but the option of doing this does not change that in Z set theories, sets do not have changing membership. > > When the universe has grown it allows bigger sets than > > where originally allowed, but the sets originally allowed are still > > sets and still the same and did not grow. How you conclude from the > > above statement that sets themselves are growing escapes me. > > It is simply a matter of definition. No, it's a matter of an axiom. MoeBlee
From: cbrown on 30 Nov 2006 14:51 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > O.K. Let A be a finite set without a largest element. > > What is the cardinal number of A. Clearly it cannot > > be any finite cardinal number? > > It cannot have a cardinal number at all. We have no cardinals in > potential infinity. Cantor knew that. > > > > Recall, you want to do more than just state that A does > > not have a cardinal number. You want to show that assuming > > that A has a cardinal number leads to a contradiction. > > Please do not mix up potential and actual infinity. Actual infinity is > required, e.g., for the real numbers. They are not potentially infinite > series, but actually infinite series. Consider just the paths in the > infinite binary tree. In this case I have shown a contradiction. Here > it is: > > The binary tree > > Consider a binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1 as binary strings. > The edges (like a, b, and c below) connect the nodes, i.e., the binary > digits 0 or 1. > > 0. > /a \ > 0 1 > /b \c / \ > 0 1 0 1 > .......................... > > The set of edges is countable, because we can enumerate them. Now we > set up a relation between paths and edges. > Relate edge a to all paths > which begin with 0.0. Relate edge b to all paths which begin with 0.00 > and relate edge c to all paths which begin with 0.01. So your relation is supposed to be a function mapping edges -> sets of paths, correct? > Half of edge a is > inherited by all paths which begin with 0.00, the other half of edge a > is inherited by all paths which begin with 0.01. Why is only 1/2 "inherited"? Why not 1, or 1/3, or any other number? Isn't "inheriting" a different function than the original relation (which was presumably a function from edges to sets of paths)? How is it that now we have a function mapping (paths X edges) -> Q? > Continuing in this > manner in infinity, we see by the infinite recursion > > f(n+1) = 1 + f(n)/2 > > with f(1) = 1 that for n --> oo > > 1 + 1/2 + 1/ 4 + ... = 2 > > edges are related to every single infinite path which are not related > to any other path. Your notation is confusing. As you have described it, edges are not related to paths; they are related to /sets/ of paths. The best I can understand by "edge e is related to path p" is that p is in the set of paths related to e; but clearly for any path p of length greater than 2, there are more than 2 edges related to p by this interpretation. To clarify, can you indicate which 2 edges are "related" to the path which represents the real number 1/3, which are not related to any other path? Cheers - Chas
From: Virgil on 30 Nov 2006 16:19 In article <1164876268.906310.52710(a)l12g2000cwl.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > The largest element of a set of numbers > today is not the same numbers as the largest element of the set > tomorrow. But the object "largest element" considered as a variable, in > fact would grow. Thatis fantasy, not mathematics.
From: Virgil on 30 Nov 2006 16:25
In article <1164877270.540944.131100(a)14g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > But why do you think that there is a > > > > > bijection N <--> Q? > > > > > > > > The reason I believe it's a theorem of Z set theory that there is a > > > > bijection between the set of natural numbers and the set of ordered > > > > pairs of natural numbers is because I've studied at least one proof. > > > > > > It is believed to be a proof. But it isn't a proof. It is demonstarted > > > for few symbols. > > > > If it is not a proof, or rather if the bijection does not exist in ZF of > > NBG or same other set theory, then WM should be able to produce a > > counterexample. > > I did. Take the natural numbers between [e*10^10^100] and > [pi*10^10^100]. You do not even know how many there are. But you insist > to prove something for them all. When I say that x <--> (x,x) is a bijection between members of any set S and the set {(x,x): x in S}, it is entirely immaterial what set S is. > > > > Particularly when WM acknowledges himself not to be a mathematician, and > > mathematicians claim otherwise? > > I am not a set theorist. Nor a mathematician of any sort. > > > > WM does not get it. In mathematics, claims unsupported by mathematically > > valid proofs do not persuade, particularly when opposed by > > mathematically valid proofs of their falsehood. > > Unfortunately today "mathematically valid" is frequently mistaken with > "matheologically believed". Not by the mathematically competent, of which WM is not one. |