Cantor Confusion
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From: David Marcus on 6 Nov 2006 13:30 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > Using Kunen's definitions, then ZFC proves that if n is a natural > > number, then n < omega. Do you disagree that ZFC proves this? > > No. Therefore omega is larger than any finite natural number. > Every natural number is finite. > omega is not finite. > > Omega is the number of digits of the diagonal of > > 1 > 12 > 123 > ... > > The number of digits of the first and any other column is omega. The > number of digits of any line is less than omega. > > The length of the matrix is omega, as a maximum taken. > The width of the matrix is omega, as a supremum not taken. > > This small problem contradicts ZFC. Before we look at your argument, please clarify the meaning of "contradicts ZFC". Which of the following do you mean? 1. "Contradicts ZFC" = "ZFC proves both P and not P, for some P". 2. "Contradicts ZFC" = "ZFC proves P. I prove not P using a different mathematical/logical system/reasoning". -- David Marcus
From: David Marcus on 6 Nov 2006 13:36 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > For the diagonal (d_kk) of the list > > > > > > 0 > > > 1 2 > > > 3 4 5 > > > 6 7 8 9 > > > ... > > > > > > we have the following mappings: > > > d_kk --> d_mk and d_kk --> d_km with k, m eps N. > > > > Normally, the word "mapping" means function. If you are trying to define > > functions, it isn't clear to me what you mean since m appears on the > > right, but not the left. > > > > Is d_mk the kth element in the mth row? > > Yes. That is the usual notation. > > > > It is curious that the set of terms of (d_km) has omega as a maximum > > > for every fixed m eps N while the set of terms of (d_mk) has omega not > > > as a maximum for every fixed m eps N. > > > > For m in N, you appear to be considering two sets: > > > > A = {d_km | k in N}, > > B = {d_mk | k in N}. > > > > If we take m = 0, then > > > > A = {d_00, d_10, d_20, ...} = {0,1,3,6,...}, > > This set has omega elements. > > > B = {d_00, d_01, d_12, ...} = {0, undefined, undefined, ...}. > > B = {0} > > Take m = 2, for instance: > > Then B' = {3,4,5} > > > Regardless, no set of natural numbers has omega as a maximum. I've no > > idea why you think your list shows that omega is the maximum of a set of > > natural numbers. > > What do you think is the difference between the first column A and any > line B of the above matrix, concerning the number of elements? Is there > a difference? If we let B_j be the set of numbers in line j, and we start labeling the lines at zero, then |A| = aleph_0, |B_j| = j + 1, for j in N. So, for any natural number j, |B_j| < |A|. Now what? -- David Marcus
From: David Marcus on 6 Nov 2006 14:39 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > Virgil schrieb: > > > > > > > In article <1162470874.593282.36250(a)b28g2000cwb.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > WM merely repeated his automatic error several more times here. > > > > > > > > WM claims that a list in which the nth listed element is a string of > > > > length at least n characters cannot produce a diagonal of length > > > > greater that any finite number of characters. > > > > > > > > > > The diagonal needs an element from every line, the n-th element from > > > the n-th line. Therefore it cannot be longer than every line. > > > > By "it cannot be longer than every line", do you mean its length can't > > be greater than the sup of the lengths of the lines? > > Its length can't be the length of a column, i.e. omega, if the width of > the matrix has only the supremum omega. All I asked was what "[the diagonal] cannot be longer than every line" meant. Was what you wrote supposed to be an answer to this question? > There is a difference between maximum and supremum which cannot be > bridged. -- David Marcus
From: mueckenh on 6 Nov 2006 15:31 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. 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"? > 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. Regards, WM
From: mueckenh on 6 Nov 2006 15:36
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. > > > > Whatever your point, you won't be able to show that merely dropping the > > > axiom of infinity from the Z axioms entails that there are only finite > > > sets. > > > > 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. > > To put it in other words: It simply an imbecile nonsense to talk about > > finished infinity without explicitly stating that it was not. > > That what was not? That it was not imbecile to talk about finished infinites. > > Modern mathematics does not use the term "finished infinity". If you > wish to use it, please define 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. Regards, WM |