Cantor Confusion
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From: Eckard Blumschein on 4 Dec 2006 13:37 On 12/1/2006 8:27 PM, Virgil wrote: > In article <45700481.7010300(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/30/2006 1:38 PM, Bob Kolker wrote: >> > Eckard Blumschein wrote: >> > >> >> >> >> Really? Fraenkel, 2nd ed. 1923, approved Cantor having managed not just >> >> to battle but also refute an assertion by Gauss. >> >> Is there any evidence for this proud claim? No. >> >> Whenever Cantor declared mathematicians like Aristotele, Locke, >> > >> > Locke and Aristotle were NOT mathematicians. Aristotle was at most, a >> > logician or an ethicist or a political "scientist" or a literarary critic. >> > >> > As a scientist he was a failure. Why? He didn't check. >> > >> > Bob Kolker >> >> Weren't Aristotele, Galilei, Newton, Leibniz, Locke, Spinoza, Peirce, >> Poincar�, Einstein, Weyl so called universal scientists? >> Fermat who paved the way for calculus was not a mathematician but a >> lawyer. Borel was a politician. > > One does not have to be a professional mathematician to be a > mathematician, and whatever Fermat and Borel were professionally, they > were also, at least as supremely competent amateurs, mathematicians. Incidentally, Leibniz perhaps contibuted most to science by his widespread correspondence. He made Guericke's discovery of electric phenomena public which led to so called Leyden bottles. >> You are perhaps a mathematician. Do you believe to understand >> mathematics deeper than one out of the mentioned? >> >> Why should mathematics be esoteric? > > Not all of it is. Various bits of it come at various levels of > abstraction, and even children understand the least esoteric bits. For my feeling, Dedekind and Cantor were lacking power of abstraction.
From: Eckard Blumschein on 4 Dec 2006 13:40 On 12/1/2006 8:20 PM, Virgil wrote: > In article <1164967792.130794.251330(a)j72g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: >> May be if you apply your personal definition of potentially infinity, >> but not if you apply the generally accepted definition. > > What "generally accepted" meaning is that? Most mathematicians do not > accept that a set can be "potentially" infinite without being actually > so. I see it quite differently: Potentially and actually infinite points of view mutually exclude each other as do countable and uncountable, rational and irrational.
From: Lester Zick on 4 Dec 2006 13:49 On Mon, 04 Dec 2006 13:08:32 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Eckard Blumschein wrote: > >> >> >> At least, a spectator is not blind. > >Neither are the players. And in addition the players know something, and >you don't. Ooooooooh!!! You tellim, oh tool and die maker of truth! ~v~~
From: cbrown on 4 Dec 2006 14:04 mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > > So your relation is supposed to be a function mapping edges -> sets of > > > > paths, correct? > > > > > > Yes. But the mapping is not the usual one (one edge --> one path). The > > > edges are subdivided in shares. > > > > Then your eventual claim that you are creating a bijection between > > edges and paths (or sets of paths) is false; you are instead showing > > that there is some mapping between "subdivisions of edges" and paths. > > That's fine; but it says nothing about whether there are equal > > cardinalities of edges and paths. > > I think, nobody would oppose to dividing the edges merely in two halves > each. If the series 1 + 1/2 + 1/4 + ... yields 2, then we can extend > this knowledge to bijections too. If I say that the sets {a} and {p,q} have the same cardinality, because I can "divide" a into 2 pieces and map them to p and q respectively, someone might well "oppose" this division as not being a bijection between the elements of the two sets. Furthermore, many sets of things (for example, the set of all finite simple abelian groups) have elements which cannot be "divided" in the way you seem to imply. What is "half" of the group of order 7? > > > > We have a function f mapping the > > > paths onto the real numbers of he interval [0,1]. The domain of f is > > > the set of all infinite paths. The function is not injective because > > > some real numbers have two representations. But the function is > > > surjective. > > > > Fine; and with a little tweaking, we can make f a bijection. We also > > have that there is a bijection h (edges -> N). You then define g so > > that for every path p, the limit of the sum over all edges e of g(p, e) > > = 2. None of this is in dispute. > > Fine. That is a big advantage for me. Usually my arguing is disputed > already at an earlier stage. > > > Hmmm. So instead of g : (paths X edges) -> Q, you have that g : (paths > > X edges X N) -> R, and that for all paths p, lim n->oo sum (over all > > edges e) g(p, e, n) = 2. > > > > Whatever. It is not disputed that this limit is 2. What is disputed is > > that /therefore/ there exists a bijection (or surjection) T : (edges -> > > paths). How do you propose to use g to construct T? > > If you dislike the fractions only... I don't dislike fractions; some of my best friends are fractions. I simply don't see that you have produced, from the functions f, g, and h, a surjective function T (edges -> paths). You appear to have no response to the request that you produce one; and instead change the problem. So I take it you agree that your original argument is flawed? Cheers - Chas
From: Virgil on 4 Dec 2006 14:35
In article <1165237393.288598.129130(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > Informally we have that a potentially infinite set is a set > > which is always finite, but to which we can add an element > > whenever we want. We say that x is an element of > > the potentially infinite set if we can add enough elements > > to get to x. > > Yes. In particular this method of adding elements guarantees that such > a set can never be uncountable. But having "added an element to it" produces a different set according to the axiom of extensionality. So EM must be working in a system in which that axiom does not hold, which means that one can never tell much of anything about what members what HE calls a set may have. > > The set N produced by induction is potentially infinite. It does not > have a largest element because it is not a fixed set but a set which > can grow. A set to which one can add an element and have the same set is not a set at all in any reasonable set theory. If WM wants such things he will have to create a separate "set" theory in which they are allowed as they are not allowed in and standard set theory/ > It has at most a temporarily largest element. For how long? > > > > How does the definition I am using differ from the > > 'commonly accepted meaning of "potentially infinite"'? > > You consider complete sets. Potential infinity is an unending process. {(n-1)/n:n in N} is , by WM;s standards, and endless process, but is completely contained in rational interval [0,1] intersection Q. > > > > The set of numbers is potentially infinite. So the real numbers > > are potentially infinite. > > So you can construct the set of all real numbers (of the interval [0, > 1] in binary representation) by: > > 0.0 > 0.1 > 0.01 > 0.11 > ... > > This set is countable. Then it appears that what Wm actually means is that WM's "potentially infinite" = our "countable" and WM's "actually infinite" = our "uncountable" |