Cantor Confusion
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From: mueckenh on 30 Oct 2006 10:01 Virgil schrieb: > > What kind of well ordering of the reals do you claim to exist? > > A well ordering in which each non-empty subset has a smallest member. That is the definition. > > > Defined? > > Catalogue? > > List? > > Else? (Please specify) > > A well ordering in which each non-empty subset has a smallest member. > > That's what well orderings are all about. I know. But the definition does not guarantee existence. In particular it does not say how the order in R differs from the order in N or Q. > > If you are asking for a rule for determining which objects come before > others, you should know that no such explicit rule is possible, but that > does not make their existence imposible. What kind of existence do you have in mind? These things do not exist unless they exist in some mind. But you say that they do not exist in any mind. So what *is* existence in this case? Regards, WM
From: Sebastian Holzmann on 30 Oct 2006 10:04 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > "dirty" is a property which without the axiom of dirt is as well > defined as "infinite" without the axiom of infinity. No. A set x (which is here to denote an element of a model M of ZF-INF) is called "finite" if x satisfies one of the following conditions: 0: x does not have an element 1: x has exactly one element 2: x has exactly two elements and so on otherwise, x is called "infinite". Where do I need the axiom of infinity to do this? >> Learn about logic, model theory and set theory (preferably in that >> order). You might be surprised. > > "Given any model of ZF-INF, one cannot be sure if there are infinite > sets" is as silly an argument as talking about sets of rats or > rabbits, or sets being clean or dirty. Learn about logic, model theory and set theory (preferably in that order). You might be surprised.
From: Han de Bruijn on 30 Oct 2006 10:19 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>I'm DOING mathematics. Mathematics is NOT independent of Physics. > > 5*10^8 m/s + 5*10^8 m/s = 10*10^8 m/s There are 10 men in a room. Each has a body temperature of 37 Celcius. This means that the temperature in the room is: 10 x 37 = 370 Celcius. Satisfied? Or do you rather prefer it in Kelvin? Han de Bruijn
From: William Hughes on 30 Oct 2006 10:25 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > There is no computable list > > > > > > of computable numbers. > > > > > > > > > > Why do you insist on this obvious fact? It does not support your > > > > > position. There is a list of all finite constructions or definitions > > > > > (encoded by numbers, Gödel). This is the definition of countablity. > > > > > The diagonal number of this list shows the listed numbers are > > > > > uncountable. Contradiction. > > > > > > > > > > > > There is indeed a list of all possible finite constructions or > > > > definitions. Call this list A. However, this list must contain things > > > > that look > > > > like definitions but are not because the method given to produce > > > > a number does not halt. We cannot get a diagonal number from A, > > > > because some of the members of A do not give numbers. > > > > So there is no contradiction. > > > > > > A contains all finite words (construction fromulas, theorems). The > > > diagonal (cannot be longer than the lines and hence) is a finite word > > > too. That is enough to obtain a contradiction. > > > > > > A also contains a number of things that look like contruction formulas > > but are not (because they don't halt). > > Forget about Turing. It is not of interest here. All sequences (words) > of A are finite / countable. > > > Thus there is no diagonal. > > The diagonal of these sequences (words) can be constructed and is > finite / countable. Piffle. Let our alphebet be {0,1,2}. Let our diagonal construction be 0->1, 1->2, 2->0. Define a finite sequence as one that has only 0's after a certain point. The set A only has sequences that have only 0's after a certain point. A begins 000... 1000... 2000... 11000... 12000... The diagonal is an unending string of ones. The set A does not contain the diagonal. > > > Thus there is no contradiction. You need a diagaonal before > > you can get a contradiction, therefore you need set B. > > Why should we not construct the diagonal of these sequeces (words) of > A? .. We can do this but the diagonal is not a finite sequence, so it is not a member of A. - William Hughes
From: William Hughes on 30 Oct 2006 10:31
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > So there is no largest B, so there is no single B. A potentially > > infinite > > set is not finite. > > There is no largest B. But every B is finite. A potentially infinite > set is always finite. Piffle. if a set A is finite then there is a single B. If a set A is potentially infinite then there is no single B. A potentially infinite set is not finite. > > > > > > > > Try to find an example for the opposite assertion: Try to construct an > > > actually infinite set with only finite numbers (which differ by a > > > constant value at least). Then you will know it. > > > > > > > The natural numbers. As you have pointed out this is a potentially > > infinite > > set so it is not a finite set. > > It is unbounded but always finite. > It is potentially infinite, therefore it is not finite. - William Hughes |