Cantor Confusion
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From: David Marcus on 20 Nov 2006 12:23 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > For any set S, B := { <x,x> | x e S} defines a bijection from S to S. > > > > > But there are more than can be > > > treated (= included in any proof) other than by induction. > > > > Not in ZF. > > > > > > > That is not proven from the mere existence of the set > > > > > > > > It _is_ proven using admissible techniques. > > > > > > Who admitted? > > > > ZF > > > > > It is proven by postulating that it is proven. > > > > It is provable from the axioms of ZF. Nothing further is needed. > > How then can countable models of ZF exist? Probably, because a "model" in this sense is not what you think it is. Before I answer your question, would you mind explaining what you understand by "countable model"? -- David Marcus
From: David Marcus on 20 Nov 2006 12:28 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > and using > > > > the existence of such an ordering. > > > > > > The existence of a well-order does not guarantee the constructibility > > > or definability of a real ordering. > > > The existence of a set does not guarantee the existence or definability > > > of a bijection or an identity mapping. > > > > No. > > > > To say that a set N exists is to say that all elements n of N exist. > > Thus the mapping n ->n for all elements n of N exists. > > That is exaggerated. There are models of ZFC (including countably many > elements a part of which could be interpreted as the set of natural > numbers) where no mapping on N exists. Can you give an example of such a model? > Why should it fail if you were right? -- David Marcus
From: David Marcus on 20 Nov 2006 12:33 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > I cannot understand your explanation given there. If you say "The > > > cardinality of omega is |omega| not omega", so you must have had in > > > mind |omega| =/= omega, > > > > No Way! If you want to misapprehend me do so, but don't confuse your > > misapprehensions with theorems of set theory. > > You wrote: "The cardinality of omega is |omega| not omega." > > Shall this sentence of yours express a difference between |omega| and > omega or not? (Now I recognize why it is so difficult to convince the > proponents sof set theory.) Franziska explained that what he meant was that the notation for the "cardinality of omega" is "|omega|", not "omega". It turns out (using a standard definition for cardinality) that |omega| = omega. -- David Marcus
From: David Marcus on 20 Nov 2006 12:36 mueckenh(a)rz.fh-augsburg.de wrote: > No. A well-order of the real numbers is not definable. A well-order of > the real numbers cannot be given by a list. There is no other means > which could well-order the real numbers. To believe that it exists > (where should it exist?) is a certificate of distinguished madness at > an advanced level. Why do things that "exist" have to exist "somewhere"? Where does the number five exist? -- David Marcus
From: stephen on 20 Nov 2006 13:43
David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> No. A well-order of the real numbers is not definable. A well-order of >> the real numbers cannot be given by a list. There is no other means >> which could well-order the real numbers. To believe that it exists >> (where should it exist?) is a certificate of distinguished madness at >> an advanced level. > Why do things that "exist" have to exist "somewhere"? Where does the > number five exist? Right next to the number four? :) Stephen |