Cantor Confusion
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From: David Marcus on 19 Nov 2006 11:32 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > The axiom of infinity does not say anything about ordinal or > > cardinal numbers. However, given that the set N exists and > > the defnition of ordinal and cardinal numbers, it is easy to > > see that if N exists it must have both an ordinal and a cardinal > > number. > > No. You assume the possibility of a bijection of the set with itself. > That is not proven from the mere existence of the set if we cannot > recognize or treat all of its elements. And even if we could, the > axiom of infinity does not prove that infinite ordinal and cardinal > numbers are numbers, i.e., that they stand in trichotomy with each > other. Are you saying that the following is not a bijection of the natural numbers to themselves? 1 <-> 1 2 <-> 2 3 <-> 3 .... -- David Marcus
From: Franziska Neugebauer on 19 Nov 2006 11:40 David Marcus wrote: > Franziska Neugebauer wrote: > >> I do not speculate about what is describable and what not. If you >> want to posit a definition do so! >> >> It is curious that you obviously don't like to give precise >> definitions of certain notions even when you have explicitly been >> asked for. > > You've only been here a short time, so you probably missed my > discussion with WM about what the word "definition" means. I had similar discussions with WM. > His definition of the word "definition" is very different from ours. > To him, a "definition" is any sort of discussion/explanation. He > denies that our sort of definition is possible or reasonable. Interestingly in WM's world there are "wrong definitions": It seems that by some magic a truth value has been attached to a definition. This is in contrast to the meaning of "wrong definition" commonly refered to. F. N. -- xyz
From: David Marcus on 19 Nov 2006 12:01 Franziska Neugebauer wrote: > David Marcus wrote: > > Franziska Neugebauer wrote: > > You've only been here a short time, so you probably missed my > > discussion with WM about what the word "definition" means. > > I had similar discussions with WM. > > > His definition of the word "definition" is very different from ours. > > To him, a "definition" is any sort of discussion/explanation. He > > denies that our sort of definition is possible or reasonable. > > Interestingly in WM's world there are "wrong definitions": It seems that > by some magic a truth value has been attached to a definition. Definition -> Explanation Wrong Definition -> Wrong Explanation > This is > in contrast to the meaning of "wrong definition" commonly refered to. I wonder what WM thinks the word "wrong" means. -- David Marcus
From: Virgil on 19 Nov 2006 15:58 In article <1163938774.893086.200910(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1163856451.247199.105160(a)h54g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Lester Zick schrieb: > > > > > > > >Sqrt(2) does exist as the diagonal of the square. But I call that an > > > > >idea in order to distinguish it from numbers which can be written in > > > > >lists and can be subject to a diagonal proof. I call only those > > > > >entities numbers which can be put in trichotomy with each other. > > > > > > > > I don't know what a diagonal proof and trichotomy may be. > > > > > > Excuse me. > > > 1) The diagonal proof by Cantor shows that any list of real numbers is > > > incomplete. For this purpose we use a list (= injective sequence) of > > > real numbers and exchange the n-th digit of the n-th number. The > > > changed digits put together yield a real number which differs from each > > > list entry at least at one position (it differs at position n from the > > > n-th list entry). This proof requires that all digits of numbers like > > > sqrt(2) do exist and can be exchanged. That assumption is wrong. > > > > WRONG AGAIN! Cantor's "diagonal" proof merely requires that the nth > > number in the list have, in principle even if not in practice, a > > determinable nth digit, which is quite a different issue. > > If not all digits of the diagonal number exist, then the diagonal > number is not an example for a number which exists but is not in the > list. But there is nothing that prevents any digit from existing, provided only that the nth digit of the nth listed number is, at least in theory, determinable. > > > > For example, it is definitely the case that for any given position n, > > the nth digit of sqrt(2) is, at least in principle, determinable, > > however impractical it might be to carry out such a determination. > > No, it is clear for everyone who is not a fanatic that in principle and > in praxis not every digit of sqrt(2) is determinable. I did not say "every", I said "any". Does WM claim that there is ANY digit in the decimal expansion of sqrt(2) that is not, at least theoretically, determinable? If so he is a fool.
From: Virgil on 19 Nov 2006 16:11
In article <1163939610.345558.62450(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > >> > The cardinality of omega is omega. > > >> > > >> The cardinality of omega is |omega| not omega. > > > > > > Kunen's "Set Theory" defines |A| to be the least ordinal that can be > > > bijected with A. So, with this definition, |omega| = omega. > > > > You are absolutely right. > > Well learned (after all)! > > Now try to understand the next step: > > If omega exists, then |omega| =/= omega & |omega| = omega. > > Then you will have reached a higher level of understanding math than > most mathematicians. What interpretation of "|omega| =/= omega & |omega| = omega" does WM suggest that is anything but false in any set version of set theory. What may be "true" in WM's notion of a set theory, need not be, and in the above example is not, true in any standard version of set theory. One of WM's assumed counterfactuals is that an infinite quantity (the length of the diagonal of his "triangular list") cannot be greater than the length of every finite line in that infinite list. That assumption alone corrupts his "theory" fatally. |