An uncountable countable set
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From: mueckenh on 10 Jul 2006 05:45 Virgil schrieb: > In article <1152450354.585870.85210(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > But it reaches past every finite value!!! > > > > > > > > There is always a finite value past every finite value. So we are and > > > > remain sufficiently save within the domain of finit values. > > > > > > So one never reaches an end. That is what "infinite" for a sequence like > > > the naturals means, "without end". > > > > Correct. And that is valid for Cantor's list too. > > Except that with Cantor's method, one need not proceed through the list > sequencially, but can deal with them all simultaneously, which with > WM's transpositions is impossible. Wrong. In order to determine the n-th line, one must count from 1 to n. This is a sequential process of conclusions from 1 to 1+1 = 2, from 2 to 2+1 = 3, and so on. Or how do you find the n-th line? If I give you a list, it will not be enumerated, but contain as many real numbers as possible, i.e., allegedly all real numbers. Regards, WM
From: mueckenh on 10 Jul 2006 05:49 Virgil schrieb: > And, in fact, all of the Cantor comparisons are proved simultaneously. How many parallel calculators in how many parallel universes are run by the fairies under your bed? Regards, WM
From: mueckenh on 10 Jul 2006 05:52 Virgil schrieb: > In article <1152450704.938408.69690(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > What we have said, and what is quite true, is that for every natural > > > there is a larger natural. > > > > Of course. And therefore it is impossible to exhaust all of them or to > > find a set which is larger than all naturals together. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ > > Except in such systems as ZF, ZFC and NBG where there is an axiom > requiring that such sets exist. > > And until WM or someone else, can find some internal > contradiction within one of these systems, they remain eminently usable, > and will be the most common standards on which set theories are based. There is no axiom that uncountable sets exist. In the contrary, it can be shown that, given ZFC is free of contradictions, it must have a countable model. This is a result of Skolem, who, therefore, argued that all the evidence that had been given for the existence of uncountable sets was inconclusive. Regards, WM
From: Christian Clason on 10 Jul 2006 08:29 mueckenh(a)rz.fh-augsburg.de wrote: > There is no axiom that uncountable sets exist. In the contrary, it can > be shown that, given ZFC is free of contradictions, it must have a > countable model. This is a result of Skolem, who, therefore, argued > that all the evidence that had been given for the existence of > uncountable sets was inconclusive. And yet, within V, it is easily provable that uncountable sets exist. The _logical_ conclusion is that the notion of countable and uncountable has different meaning "inside and outside V" (since the "meaning" of the element relation is also part of the model). This is known as the Skolem paradox, which is discussed in detail in most books on logic or elementary model theory. Best regards, Christian Clason
From: mueckenh on 10 Jul 2006 11:05
Christian Clason schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > There is no axiom that uncountable sets exist. In the contrary, it can > > be shown that, given ZFC is free of contradictions, it must have a > > countable model. This is a result of Skolem, who, therefore, argued > > that all the evidence that had been given for the existence of > > uncountable sets was inconclusive. > > And yet, within V, it is easily provable that uncountable sets exist. These "proofs" are wrong. At least can be sown that the opposite results can also be obtained: uncountable sets can also be proved countable. > The > _logical_ conclusion is that the notion of countable and uncountable has > different meaning "inside and outside V" (since the "meaning" of the > element relation is also part of the model). Yes, a model where the bijection cannot be defined. Nice. A bijection can be formulated with less than 100 letters. But those letters are not available in all those models? Countable infinity is not enough to supply so many letters? Ridiculous! >This is known as the Skolem > paradox, which is discussed in detail in most books on logic or elementary > model theory. Why, do you think, was Skolem an intuitionist? Why was it his opinion that uncountability was nonsense (he said "inconclusive" - at those times the habits were not yet spoiled by the internet). Regards, WM |