Cantor Confusion
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From: Virgil on 18 Nov 2006 15:06 In article <1163856776.906287.21380(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > >> > > >> http://mathworld.wolfram.com/InitialSegment.html > > >> > > >> What you call "complete initial segment" is not an *initial* segment > > >> but the whole set of lines. > > > > > > What is a name? > > > > You should rephrase what you mean. > > > > > What *counts* is this: It is asserted that the number of natural > > > numbers is omega > > > > "It" is asserted that the *set* of natural numbers is (does exist) and > > is named omega. The *cardinality* of omega is aleph_0. > > The cardinality of omega is omega. It is usual to denote it by aleph_0, > but it is allowed to denote it by omega. Even Cantor is said to have > done so ... Since WM has so often claimed that Cantor is all wrong, he is hardly now in a position to justify anything by saying that Cantor was right about it. > And there are not enough natural numbers that we could collect omega or > aleph_0 of them, although this wrong definition exists. Since omega is, by definition, such a collection, WM is henceforth barred from using it by his own argument. > We see Who is this "we" that sees thing that are no there? > > > > B := { <0, 1>, <1, 2>, <2, 3>, ... } > > > > B is an explicit bijection between the naturals (elements of omega) and > > the numbers in the first column. > > B does not include omega. But B is order isomorphic to omega. Which for ordinality is just as good. > If omega were only the fact that this > bijection does include all natural numbers, then you had no problem > with B. But if omega is considered a number which even can be > increased, Does WM claim that there are no ordinals of which omega is a proper subset (where it is understood that every ordinal is the set of all previous ordinals)? > you must assume that there are > more natural numbers d_nn than natural numbers n. Not unless your brain is cross wired. N need never be a proper subset, or proper superset, of N.
From: Virgil on 18 Nov 2006 15:11 In article <1163856869.641128.249070(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > > > mueckenh wrote: > > > without axioms, yes. For instance: I + I = II (after translating "+" > > > and "="). Therefore I call this an absolute truth. > > > > Which axioms are you using to describe the "+" and "=" operators? > > Axioms? For which purpose? Perhaps "definitions and axioms" wold have been better. > Do you think the symbols constituting the > words constituting the axioms are easier or clearer to understand than > the symbols "+" and "="? Take an apple and then another apple. Show > the apples first apart and then together. Repeat with oranges or > fingers or mixed objects, possibly. That defines all that is needed. No number of specific cases ever "defines" a general rule sufficiently for purposes of mathematics.
From: David Marcus on 18 Nov 2006 15:14 Franziska Neugebauer wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > What *counts* is this: It is asserted that the number of natural > >> > numbers is omega > >> > >> "It" is asserted that the *set* of natural numbers is (does exist) > >> and is named omega. The *cardinality* of omega is aleph_0. > > > > 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. > > although this wrong definition exists. > > You have already been informed about this misconception of yours. I fear that WM thinks he is informing us of our misconceptions. -- David Marcus
From: Virgil on 18 Nov 2006 15:15 In article <1163857371.306845.100530(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > I claim case iia: There is a potentially infinite sequence N = > > > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > > > treat all of its elements. In particular we can never complete this > > > set. We can never put it into a list > > > > OK. Knock yourself out. > > I read this several times from you. What does it mean? It means that you in waters over your head and can't swim. > > > Note however that this case > > is not consistent with assuming the axiom of infinity. > > The axiom of infinity says that the set N exists. > > But it does not say that it has an ordinal number and a cardinal > number. Because "ordinal number" and "cardinal number" have not yet been given definitions. Those definitions, and others, follow from the axioms rather than preceding them. WM again has cart-before-horse-itis.
From: David Marcus on 18 Nov 2006 15:26
imaginatorium(a)despammed.com wrote: > > Dik T. Winter wrote: > > In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > In > > > > principle no axiom is necessary. But you need a few to have some start > > > > to work with. > > > > > > That's the question. By means of axioms you can produce conditional > > > truth at most. I am interested in absolute truth. Axioms will not help > > > us to find it. I don't think we need any axioms. > > > > If you want to find absolute truth you should not look at mathematics. > > Really? There are two groups of order 4; could any truth be more > absolute than that? I think it depends on what the words mean. If the axioms are correct in your model, then the theorems are correct in your model. However, this seems to only be relative truth, i.e., if the axioms are "absolutely true", then the theorem is "absolutely true". Why did you pick the statement you did, rather than something like 2 + 2 = 4? -- David Marcus |