Cantor Confusion
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From: mueckenh on 11 Nov 2006 16:54 MoeBlee schrieb: > > p. 118, considering that the set of all sets does not exist, they > > write: a reasonable way to make this conform to a 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". > > > > Why should they write "reasonable" if they found it unreasonable? > > Because, as YOU SKIPPED RESPONDING to my explanation for you, and as > would be clear had you bothered to read and UNDERSTAND the remarks in > that section of the book, theyr'e NOT claiming that sets in general > grow or are indeterminate. They are pointing out that if we adopt > different axioms, then the entire universe of all sets is something > different depending on which axioms we adopt. I will not discuss this topic further. Only a last remark for the lurkers: In fact, there are axiom systems which tolerate the set of all sets. Therefore, if different axiomatic systems were discussed by Fraenkel et al., no capability of "growing" was required at all. Further there is no talk about "something different" sets but about growing sets. Therefore exactly this growing of sets is meant as a reasonable view. Unfortunately MoeBlee is far from understanding anything behind his formalism. "The formalists are like a watchmaker who is so absorbed in making his watches look pretty that he has forgotten their purpose of telling the time, and has therefore omitted to insert any works." (Bertrand Russell) .. Regards, WM
From: mueckenh on 11 Nov 2006 16:56 MoeBlee schrieb: > So let's be definite here. Do you have a strong belief that your > argument can be expressed in the language of set theory and uses only > first order logic applied to the axioms of set theory? I need no belief. If all consistent mathematics can be expressed in ZFC, then my argument can be expressed in ZFC too. Regards, WM
From: mueckenh on 11 Nov 2006 17:04 David Marcus schrieb: > > You carried out a survey? Or is that your guess? I know some > > mathematicians who know what a mathematician should understand by > > "limit", "number", "grow". > > Please post their definitions of the words. Try to get a clue of Cantor's second creation principle. Then you will know these things. > > > > Your changing "finished entity" to "finished infinity" is a bit of a > > > stretch. Regardless, Hrbacek and Jech are clearly making a philosophical > > > comment, not discussing mathematics. > > > > They were discussing mathematics. Philosophy belongs to mathematics. > > That's nonsense. That's ignorance. > > > > So, it remains true that "finished > > > infinity" does not have a mathematical meaning. Ironically you are even right, for once, though you don't know it. > > > term in a mathematical discussion, it is incumbent on you to define it. > > > > This term has been used in a mathematical discussion by Hrbacek and > > Jech. I don't see why I shouldn't do the same. > > Several reasons: They were discussing philosophy of mathematics, not > mathematics itself. You don't understand what they were saying. They said: "Some mathematicians object to the Axiom of Infinity on the grounds that a collection of objects produced by an infinite process (such as N) should not be treated as a finished entity." > And, you > haven't given a definition of the term. I thought you'd know transitivity: Take the expression "finshed entity" where entity is a variable like "the set X" in set theory. Now replace this variable by a fixed set like N, which in mathematics, is an infinite process. This leads to "finished infinite process", abbreviated by "finished infinity". Was this simple enough for you to understand? >You can use any term you wish, > but only if you define it. That is one of the rules of the game. Another rule is that every player should at least know 10 terms, including the application of transitivity. > If you > don't want to play, then don't, but you can't unilaterally change the > rules. > > > Which mathematics do you allude to? > > That which is taught in school, explained in textbooks, published in > journals, and discussed by some in this newsgroup. I teach at a school, I wrote textbooks (and my new book is in print), I published in journals and I wrote in newsgroups. Regards, WM
From: mueckenh on 11 Nov 2006 17:06 Virgil schrieb: > > They were discussing mathematics. Philosophy belongs to mathematics. > > There may be some benighted mathematicians who claim so, but there are > as many philosophers, and others, who will claim that mathematics > belongs to philosophy. Did I say that *all* philosophy belongs to mathematics? The philosophy of mathematics belongs to mathematics. Regards, WM
From: mueckenh on 11 Nov 2006 17:08
David Marcus schrieb: > William Hughes wrote: > > > a: Any infinite set of numbers must contain an infinite number > > > > b: It is possible to have an infinite set of numbers that does > > not contain an infinite number. > > WM, please tell us if you agree or disagree with the statements a and b > above. You will not understand it. But I'll try. The potentially infinite set N does not contain an infinite number. In this case (b) is correct. An actually infinite set is a set which has a cardinal number like omega. In set theory every set is actually infinite. So N is actually infinite. If its cardinal number omega is claimed to count the members of N, then there is a contradiction, because omega counts all n in N but does not belong to N while the initial segments of N are counted by natural numbers. This statement remains true as far as the natural numbers reach, i.e., for all natural numbers , i.e., for N. The contradiction becomes obvious in the matrix discussed or in the following few lines: Cantors first approach read: n+1, n+2, n+3, ..., 1, 2, 3, ..., n is a sequence with ordinal number omega + n. Here omega does not appear in the sequence. omega is nothing but the sequence n+1, n+2, n+3, .... But Cantor wanted omega to be an ordinal number alike the finite ordinals. Therefore, omega must appear in the sequence: n+1, n+2, n+3, ..., omega, 1, 2, 3, ..., n What is the ordinal number of this sequence? Regards, WM |