Cantor Confusion
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From: William Hughes on 22 Nov 2006 10:53 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > The countability of the set W of all the words of a finite alphabet is > > > proved by other means than the usual countability proof. The latter > > > consists in constructing a bijection with N, i.e., in constructing a > > > list. But it is impossible to construct a list of all words. It is > > > impossible to construct a bijection between W and N. Why should it be > > > possible to construct a bijection between N and N? > > > > Because a bijection between W and N cannot be the identity map > > but a bijection between N and N can be the identity map. > > If both, W and N, are countable, then renaming the elements of one of > them leads to an identity map. You are confusing bijections within the model to bijections outside of the model. - William Hughes
From: mueckenh on 22 Nov 2006 10:55 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > No. You can develop "potential set theory". Just define > > > an element of a potential set to be an element of any one > > > of the arbitrary sets that can be produced. Now go through > > > set theory and add the word "potential" in front of each > > > occurence of the word set. There is no difference between > > > saying "a potentially infinite set exists" and saying "an actually > > > infinite set exists". > > > > A potentially infinite set has no cardinal number. It cannot be in > > bijection with another infinite set because it does not exist > > completely. > > > > Piffle. You need to study your potential-set theory. > > We have defined what it means to be an element of > a potential set. We never have all elements available. > > Therefore we can define bijections between potentially > infinite sets. We can never prove hat a bijection fails, like in Cantor's argument. > > Therefore we can define cardinalities of potentially infinite > sets. Yes, we can define that a set is finite or that it is infinite, denoting the latter case conveniently by oo. That's all. Regards, WM
From: William Hughes on 22 Nov 2006 10:58 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Randy Poe schrieb: > > > > > > > > > > > > > > > > You never heard of countable uncountable models? > > > > > > > > I've never heard of something which is both countable > > > > (can be bijected to N) and uncountable (can not be bijected > > > > to N). That would seem to require both P and (not P) to > > > > be true for some proposition P. > > > > > > Indeed, that is my impression too. But ZFC cannot tolerate P and notP. > > > Therefore people have dispelled it. If ZFC is correct, then, according > > > to a theorem of Skolem, it must have a countable model. But the most > > > important theorem of ZFC proves the existence of uncountable sets. > > > Therefore the worshippers of transfinity proclaim the existence of a > > > set which is countable (as Skolem requires) "from outside" but does not > > > contain the bijection with N internally. Therefore "inside the model", > > > countability cannot be proved - and all is fine. > > > > More piffle. Skolem's "paradox" does not concern us. > > It does. It is not only a paradox but an antinomy. > > > Yes there > > is a countable model of ZFC. Yes, within this model there is a > > "set" of natural numbers N', and a "set" of real numbers R' such that > > within the model there is no "bijection" between N' and R'. But what > > you claimed > > was that there was no bijection between N' and N'. > > If someone would have claimed, before 1920, that between a countable > set N' and a countable set R' no bijection can exist, you would have > answered piffle or so. Now you are in the same situation. Yes I would have replied piffle. I would have been right. The only way this can be true is if you reinterpret "countable". Do not confuse the meaning of "countable" within a nonstandard model with the meaning of "countable" in the standard interpretation. - William Hughes
From: Eckard Blumschein on 22 Nov 2006 11:06 On 11/21/2006 6:12 PM, William Hughes wrote: > Eckard Blumschein wrote: >> > (i.e. the difference between actual and potential infinity) is >> > mostly one of terminology. >> >> Yes, but I got aware that actual and potential infinity are two >> different views both directed towards the same object of natural numbers >> from different levels of abstraction. The potential infinite view >> belongs to counting and is able to separate from each other all single >> numbers. The actual infinite one provides the fiction of all natural >> numbers at a higher level of abstraction. You cannot have both views at >> a time. >> > > Why not. There is nothing about assuming the set of all natural > numbers exists that precludes counting or the ability to sepatate > from each other all single numbers. Actual infinity is not as handsome as it seems to be. > The elements of the > potentially infinite set are exactly the same elements with > exactly the same properties as the elements of the > actually infinite set. I object to exactly this assumption. Actual infinity denotes a diffent quality, not a larger quality, not a quantity at all. >> And the elusive belief hidden within and conveyed by this name. >> Aren't I correct? > > No. You can develop "potential set theory". Quite a while ago, it was a surprize to me: Axiomatic set theory does not really require the actual infinity. > Just define > an element of a potential set to be an element of any one > of the arbitrary sets that can be produced. Now go through > set theory and add the word "potential" in front of each > occurence of the word set. There is no difference between > saying "a potentially infinite set exists" and saying "an actually > infinite set exists". Yes. This is indeed a clever method of obscuration. Nonetheless, if one prefers to distinguish between rational and real numbers, then the reals are only distinguished by being fictitious and therefore uncountable. Genuine reals being really real are only required for theoretical considerations. They do however, have properties quite different from the properties of genuine numbers. This is my original concern. After I already settled the somewhat puzzling matter by means of really real numbers, I took me some time to understand the relationship to the still thaught naive set theory, axiomatic set theory and what a comprehensive understanding of mathematics really needs. Appreciating regards, Eckard Blumschein > > - William Hughes >
From: mueckenh on 22 Nov 2006 11:10
William Hughes schrieb: > It is trivially true that given any two elements of the diagonal > that a line exists that contains both elements. Fine. > > It would be quantifier dyslexia to claim that there exists > a line that contains any two elements of the diagonal. So these any two elements are different from those any two which can be in a line? How do you distinguish the first any two from the second any two? > > ["two" can of course be replace by any natural number.] No. Your quantifier magic may apply to sets of men and women and dancers and what else you may like, but not to linearly ordered sets of finite elements. In case of the finite lines your assertion is simply absurd. This implies that the diagonal has not omega elements. > Your claim is that there is a bijection with one line. > Call this line Kumquat. Kumquat has a largest element. > Since there is a bijection between the elements of Kumquat and > the d_nn, there must be a largest d_nn. > If there is no bijection with one line, then there must be an element of the diagonal outside of every line. (Because what can be done with several *finite* lines in their linear order, can be done with one line). This assumption is 2^Kumquat. But if you want to entertain that idea, do it. There is no reason to further discuss this aberration of mind and, above all, no hope to rectify it. Regards, WM |