Cantor Confusion
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From: mueckenh on 22 Nov 2006 11:14 William Hughes schrieb: > > The two statments > > i: P(n) is true for every n an element of N. > ii: P(N) is true > > are not the same (trivial example P(x) is true iff x is an element of > N). > Induction can be used to prove statements of the form i. > (eg all elements of N are finite). Induction cannot be used > to prove statements of the form ii (e.g. N is finite). Here again your mathelogy comes to the surface. N is nothing but the collection of all natural numbers. They count themselves. If all are finite, then all are finite, i.e., then N is finite. Regards, WM
From: mueckenh on 22 Nov 2006 11:15 Franziska Neugebauer schrieb: > 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. > > Is the "identical sequence" n |-> n a set, too? In ZF and ZFC everything is a set. Regards, WM
From: mueckenh on 22 Nov 2006 11:21 William Hughes schrieb: > 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. And who told you that you were outside? But you misunderstood me. W and N are both in "our" universe. Regards, WM
From: Eckard Blumschein on 22 Nov 2006 11:26 On 11/22/2006 1:46 PM, William Hughes wrote: > 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. Since I am convinced that Dedekind and Cantor were wrong, I avoid the expression cardinal number. Why do you not simply write instead: Something potentially infinite is countable because it is not thought to exhibit the impossible property to include all elements of something actually infinite? > Piffle. I recommend to avoid hurting words. >You need to study your potential-set theory. > > We have defined what it means to be an element of > a potential set. > > Therefore we can define bijections between potentially > infinite sets. If one considers something as actually infinite, then this point of view even hinders bijection. One cannot eat the cake and have it. potentially infinite and actually infinite point of view exclude each other as do discrete numbers and continuity. > Therefore we can define cardinalities of potentially infinite > sets. I feel in a position similar to that of an atheist who faces a believing child. Please forgive me my lack of belief. Yes there is Santa Claus.
From: mueckenh on 22 Nov 2006 11:27
William Hughes schrieb: > > > > > 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 There is no standard model of ZFC, so there is no non-standard model. > with the meaning of "countable" in the > standard interpretation. Countable means: There exists a bijection with N. After having recognized that this simple definition leads to a contradiction, people defined countable in and outside differently. And if they will recognize that there is really a contradiction in ZFC, then they will define contradiction differently. I am sure, they will not fail. Regards, WM |