Cantor Confusion
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From: mueckenh on 23 Nov 2006 07:23 Virgil schrieb: > In article <1164192462.824282.134620(a)m7g2000cwm.googlegroups.com>, > 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. > > Not true in any extant axiom system. In every axiom system that has been > presented so far, there are finite sets and possibly infinite sets, but > nothing in between. Something which is potentially, but not actually, > infinite cannot be a set at all in any set theory yet seen here. And just that is the reason why any set theory is waste. Regards, WM
From: mueckenh on 23 Nov 2006 07:30 Virgil schrieb: > In article <1164195797.399821.219210(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > One counter example contradicts ZFC: > > > > There is not one single element of the diagonal which is not contained > > > > in a line. This line contains this and all preceding elements. > > > > > > While true, it is irrelevant to the issue of whether there is one line > > > containing every element of the diagonal, which there is not. > > > > > > WM conflates "every element of the diagonal is in SOME line", which is > > > true, with "every element in the diagonal is in THE SAME line", which > > > is false. > > > > That means we need at least two lines for the elements o the diagonal? > > Not to anyone who understands logic. What is the opposite of "at least two"? > > What it does mean is that we need infinitely many finite lines to get > all of the infinitely many members of the diagonal. So infinite is less than two? > > At least to people of any sense. Look, what is the value of the statement of a fool that his companion is not a fool? I think, there is no value at all. > > > Please give an example which requires that at least two different lines > > are needed to contain two elements of the diagonal. > > Please give any line for which every element of the diagonal is in THAT > line. Project the diagonal in horizonat direction. Then you have it. Regards, WM
From: mueckenh on 23 Nov 2006 07:33 Virgil schrieb: > In article <1164195919.471077.233250(a)f16g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <456304B0.70705(a)et.uni-magdeburg.de>, > > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > > > > Being admittedly not very familiar with set theory, I nonetheless wonder > > > > if sets are considered like something going on forever. > > > > > > Sequences do, sets do not. > > > > > Sequences are sets. > > > > Regards, WM > > Sequences are functions, a very special sort of set, those with domain > N. > > Since not all sets are sequences, not all sets have the properties that > sequences must have in order to be sequences. Have you lost the rest of your mind? I said "all sequences are sets", but not "all sets are sequences." Regards, WM
From: Albrecht on 23 Nov 2006 07:34 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > 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. > > Piffle. > > Real numbers are represented by potentially infinite sequences. > A list of reals is a function that takes an element of the potentially > infinite > set of natural numbers and returns a potentially infinite sequence. There is no function that takes an element of the potentially (or actually) infinite sequence of natural numbers and returns a potentially (or actually) infinite sequence in that way that all possible potentially (or actually) infinite sequence are covered. But it is not because there are more infinite sequences than natural numbers. It is just because there exists no sequence of the infinite sequences. Best regards Albrecht S. Storz > The diagonal number is a potentially infinite sequence. > We show the diagonal number is not a member of the list > in exactly the same way as before. > > > > > > > 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. > > > > Defintion. B is a g-subset of a potentially > infinite set A if B is a set or a potentially infinite > set and any element of B is an element of A. We can > now contruct potentially infinite power sets, and > the usual theory follows. > > You are confusing potentially infinite with computable. > While it is true that a computable set is either > finite or potentially infinite, it is not true that a potentially > infinite set is computable. > > - William Hughes > > Regards, WM
From: mueckenh on 23 Nov 2006 07:40
Virgil schrieb: > In article <1164196369.064190.253870(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1164126395.211430.7520(a)h54g2000cwb.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Tell me which point is not accepted: > > > > 1) Every line which contains an index of the diagonal, contains all > > > > preceding indexes of the diagonal too. > > > > 2) Every index of the diagonal is in a line. > > > > 3) In order to show that there is no line containing all indexes of the > > > > diagonal, there must be found at least one index, which is in the > > > > diagonal but not in any line. > > > > > > This one is flat out false, unless one assumes, a priori, a last line > > > and a last member of the diagonal. > > > > We assume not a last line, but we assume that eery line has finitely > > many indexes. And this is true. > > What you allege in (3) does not follow from this. > It is true that given any line there will be a diagonal element not in > that line. > It is false that given a diagonal element there is no ine which contains > it. > > WM again dyslexes his quantifiers. > > > > > > It is certainly false in ZF or NBG, where such an assumption is also > > > false. > > > > For finite indexes it is correct. > > Infinite things are not constrained to behave in all respects like > finite ones. That is because "infinite" means "not finite". But *the lines are all finite*. That's just why I chose the EIT as example. > > > > > > It is quite enough to show that for every index in the diagonal, > > > except 1, there is some line not containing that index. Then no line > > > can contain every index. > > > > Name two finite indexes which cannot be in one line. > > That is not at all relevant to what I said. It is relevant since every line is finite. So we have to deal with finite indexes only. And every line has finitely many indexes. So we have to deal with finitely many indexes only, as far as the lines are concernded. (That's why you raised that silly argument with the diagonal loger than every line.) So your magic belief is easily destroyed. Regards, WM |