Cantor Confusion
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From: Virgil on 6 Nov 2006 17:40 In article <1162829332.263965.226560(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > The number of elements of a column is omega. The number of elements of > the diagonal is omega. Both are not suprema but maxima. The width of > the matrix has not the maximum omega. The width of the "matrix" has the same supremum as its diagonal and the number of elements in a column. > > > > No, there does exist a set of infinitely many columns. However, this > > set is not a natural number. It is the supremum of a set of natural > > numbers. A supremum of a set A does not have to be a member > > of A. > > But it can be. The set of lines has a maximum, namely omega. A maximum line would of necessity be a last line. The set of lines has no last line, so cannot have a "maximum", but can have a supremum. > Or does > the complete set not exist at all? It does exist , but not with the property of having a maximal member. > > > So while this set is an infinite number it is not a natural > > number. > > Why is the set of columns smaller than the set of lines? Or isn't it? Why should it be when there is a natural bijection between them, at least if there is no finite upper limit on line lengths? > > Regards, WM
From: Virgil on 6 Nov 2006 17:44 In article <1162845094.170762.321610(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > You should not try to judge about topics which you must reject because > you do not understand them. Do not, as some ungracious pastors do, Show me the steep and thorny way to heaven; Whilst, like a puff'd and reckless libertine, Himself the primrose path of dalliance treads And recks not his own read.
From: MoeBlee on 6 Nov 2006 17:46 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > > Whatever your point, you won't be able to show that merely dropping the > > > > > > axiom of rabbithood from the Z axioms entails that there are only > > > non-rabbit > > > sets. > > > > Your analogy just agrees with what I said. You don't seem to have a > > point. > > > So you would not exclude sets with rabbithood in ZFC? That is an idiotic question. If 'rabbithood' is defined in set theory, and the sentence "There does not exist a set with rabbithood" is not a theorem, then we can't claim that there are no sets that have rabbithood. But unlike 'is infinite', it doesn't seem that 'rabbithood' is defined in set theory. You are mistaken that we can't define a predicate without having first proven that there exists an object that has that predicate. > > > To put it in other words: It simply an imbecile nonsense to talk about > > > finished infinity without explicitly stating that it was not. > > > > I don't know what you mean by "stating that it was not". > > It simply an imbecile nonsense to talk about finished infinity without > explicitly stating that it was not an imbecile nonsense to do so. Ooo kay, you say so. > > Neither the axiom of infinity nor its negation are theorems of ZF > > without the axiom of infinity. Get it? > > > And infinity without an axiom stating its existence is as meaningful as > the rabbithood of sets within ZFC. No, since 'is infinite' is defined in ZFC while 'rabbithood' is not. What an idiotic argument you use. >You need not declare or prove that a > triangle is faster than a matrix when there is no axiom stating the > contrary. Again, that's idiotic. If we have defined the relation 'faster than' for triangles and matrices, then we can talk about whether a triangle is faster than a matrix. But we haven't defined 'faster than' for triangles and matrices, so we don't talk about whether a triangle is faster than a matrix. MoeBlee
From: Virgil on 6 Nov 2006 17:50 In article <1162845365.468299.53500(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > As > > > > to set theory, for the tenth time: Without the axiom of infinity it is > > > > UNDETERMINED whether every set is finite. > > > > > > For the eleventh time: Infinity is NOWHERE. To assume the existence of > > > actual infinity is one of the greatest errors of human mind. Therefore, > > > without explicitly assuming this notion, it cannot be anywhere. > > > > Moe was discussing axiomatic set theory, specifically ZF without the > > axiom of infinity. You clearly are discussing philosophy, e.g., > > "greatest errors of the human mind". If you wish to discuss philosophy, > > feel free, but please do not inject philosophy into a discussion of > > mathematics without saying you are doing so. > > The consideration of actual infinity without any axiom of infinity is > not set theory but philosophy. The consideration of whether any sort of infinite set is barred in a set theory without an axiom to guarantee its presence is sound mathematics, even though WM's grasp of mathematics is too tenuous to comprehend it. > > Modern mathematics assumes that there are sets which are larger than > infinite sets. That is the same as to say after one finished infinity > we consider the next infinity. > > > > Usually, the word "read" means read and understand. > > But that is not always correct. Consider your own readings. WM should mend his own fences in that regard before complaining about others.
From: MoeBlee on 6 Nov 2006 17:53
mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > Here is my translation: It is allowed to understand the new number > > > omega as limit to which the (natural) numbers n grow, if by that we > > > understand nothing else than: omega shall be the first whole number > > > which follows upon all numbers n, i.e., which is to be called larger > > > than each of the numbers n. > > > > That's not a definition. It is just a remark. > > It is a definition. You should not try to judge about topics which you > must reject because you do not understand them. Please learn: A > definition is an explanation in other words. Nothing more can be done. No, YOU are IGNORANT of the subject of mathematical defintions. A mathematical definition is NOT just any explaination in words. > Would you understand: "The new number omega is the limit to which the > (natural) numbers n grow"? Or is even hat too incomprehensible to you, > because you don't know the meanings of "number", "limit", and "grow"? No, it's just that you haven't given MATHEMATICAL definitions of them. Just because there are non-formal meanings of these words doesn't entail that they come with MATHEMATICAL definitions all attached and ready for use. > > All it says in modern > > terminology is that omega is the smallest ordinal larger than every > > natural number. No one disagrees with this. > > > > > > But I'd like to know what you might > > > > propose as a mathematical definition of 1/omega in Z set theory. > > > > Notice, I'm not asking what Cantor wrote. I'm asking what is the > > > > definition of 1/omega specifically in Z set theory. > > > > > > If omega is larger than 2, then 1/omega is smaller than 1/2. > > > > You were asked for a definition of "1/omega" and you state something > > that is not a definition and which is meaningless without a definition > > of "1/omega". > > > > > Now replace 2 by n and let n-->oo. > > > > Still meaningless without a definition of "1/omega". > > You should not try to judge about topics which you must reject because > you do not understand them. We reject mere undefined (and not stated to be primitive) verbiage thrown around as if it is mathematics. Meanwhile, you reject set theory as you do not understand it, as you have never bothered to learn it. MoeBlee |