Cantor Confusion
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From: William Hughes on 23 Nov 2006 09:17 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > If there is no bijection with one line, then there must be an element > > > of the diagonal outside of every line. > > > > No. Not unless the one line > > contains every element from every line. > > This holds for every finite line. Every finite line contains every > element from every preceding line. There are only finite lines. I claim There does not exists a line L_1 such that L_1 contains every element from every line. You counter with For every line L_1, L_1 contains every element from every line preceding L_1. However, these two statments are not contradictory. "every element from every line" is not the same thing as "every element from every line preceding L_1" Yes it is true that For any element of the diagonal, d_nn, there exists a line L_2, such that L_2 contains d_nn. However, the statement There exists a line L_1, such that L_1 contains every element from every line preceding L_1. is not the same as the statement There exists a line L_1, such that L_1 contains every element from every line preceding L_2. So we cannot use There exists a line L_1, such that L_1 contains every element from every line preceding L_1. to show that There exists a line L_1, such that L_1 contains every element from every line - William Hughes
From: William Hughes on 23 Nov 2006 09:22 Albrecht wrote: > 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. > Since you have no support for that last statement, the entire argument holds no weight. - William Hughes P.S. Any progress on the definition of complete?
From: Albrecht on 23 Nov 2006 10:33 On 23 Nov., 15:22, "William Hughes" <wpihug...(a)hotmail.com> wrote: > Albrecht wrote: > > William Hughes schrieb: > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > William Hughes schrieb: > > > > > > mueck...(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. > > > Since you have no support for that last statement, the entire > argument holds no weight. But it is already proved. The diagonal argument of G.Cantor shows exact this fact. Just use the only meaningful axiom of infinity: all infinite collectivities have one aspect in common: they are infinite, endless, unfinishable. > > - William Hughes > > P.S. Any progress on the definition of complete I work on it. In the meantime I use yours. Best regards Albrecht S. Storz
From: Eckard Blumschein on 23 Nov 2006 12:05 On 11/23/2006 9:14 AM, Virgil wrote: >> Eckard>> That little ambiguity? Doesn't it make a categorical difference >> if a set >> >> has been a priori set for good? >> > >> Virgil> When one speaks of the sequence {1,2,3,...} and another speaks >> of the >> > set {1,2,3,...}, the notational ambiguity should not override the words >> > "sequence" and "set". >> >> Perhaps you did not get or at least did not accept my point: >> Does it not make a categorical difference whether a set is something >> unchanging beeing already perfect for good or something that just >> formally includes the vital property of the series of nagtural numbers >> to have no end at all? >> The notation {1, 2, 3, ...} is ambiguous with this respect. It depends >> on the decision whether ... denote potential or actual infinity. > > Are the labels "sequence" and "set" ambiguous to EB? If so perhaps he is > just muddled. The ambiguity resides within the three points "..." They may denote either actual or potential infinity. >> Meanwhile I see mounting evidence for my suspicion that set theory is >> some sort of (self?)-deception. > > EB sees what he tells himself he should see, regardless of whether there > is anything there to see of not. Those who follow the discussion will not immediately change their opinion but should have a chance for doing so.
From: Eckard Blumschein on 23 Nov 2006 12:07
On 11/23/2006 9:16 AM, Virgil wrote: > In article <456556CD.4030107(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > >> Virgil> While all sequences are sets ( as functions from N to some set of >> > values) not all sets are sequences. >> >> So you could be correct if referring to sets that are no sequences? >> Look above: I referred to something going on forever. > > Such as EB's nonsensical utterances about that of which he displays such > great ignorance? Sorry, the factual question of concern is perhaps meanwhile missing, replaced by nonsense and ignorance. Let's stop now. |