Cantor Confusion
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From: mueckenh on 6 Feb 2007 04:53 On 4 Feb., 19:47, Fuckwit <nomail(a)invalid> wrote: > On 4 Feb 2007 10:39:08 -0800, mueck...(a)rz.fh-augsburg.de wrote: > > > > >>> What is 4 in mathematics? > > >> In axiomatic set theory it's (usually) the set > > >> {0, 1, 2, 3}. > > > What is 3, what is 2, what is 1, what is 0, what is > > the empty set? > > The empty set is the only set which does not have any elements also streng genommen als solche gar nicht vorhanden ist (Cantor) > Its > existence (in ZFC) is guaranteed by the axiom of subsets (Aussonder- > ung). You know, modern math is based on _axioms_, not some sort of > hand waving (you seem to prefer). Usually the empty set is denoted > with "{}". The existence of the empty set is not at all guaranteed. There is an axiom which requires the existsence of the non existing and seems to make some people happy (like the axiom which requires the finity of the infinite). > > Then we may define (in an entirely non-circular way): > > 0 := {} > 1 := {0} > 2 := {0,1} > 3 := {0,1,2} > 4 := {0,1,2,3} > > > > > That is set theory, not mathematics. > > Well, the last time I've checked it set theory was mathematics. I don't believe that you are able to check what mathematics is. Regards, WM
From: mueckenh on 6 Feb 2007 05:02 On 4 Feb., 20:28, Virgil <vir...(a)comcast.net> wrote: > > > Not so. Induction can only prove every set of naturals which is bounded > > > above by a natural is finite, but that does not, according to the axiom > > > of infinity, exhaust the possible sets of naturals. > > > There is no natural number in this set which is not covered by > > induction. So, which number is missing to exhaust N? > > Who said anything like that? WM misreads so consistently that he must be > doing it intentionally. You say: ... but that [Induction] does not, according to the axiom of infinity, exhaust the possible sets of naturals. So what remains to be exhausted wich is not exhausted by induction? > > > > > > What Induction says is precisely: > > > Given a set, S, of natural numbers such that > > > (a) S contains the first (smallest) natural, and > > > (b) whenever a particular natural is a member of S, > > > the successor of that natural is also a member of S, > > > then every natural is a member of S. > > > > And induction says nothing else. > > > Induction says: If P holds for n then it holds for n+1. > > On the contrary, it says no such thing. It is entirely up to the person > attempting to use induction to PROVE that if P holds for n then it holds > for n+1. There is nothing in induction to require it. If "If P holds for n then it holds for n+1" is satisfied, then the set is infinite. Otherwise it is finite. > The only esoteric version of infinite (not finite) is WM's. > In standard set theory, a set is either finite or not finite with no > third option. In set theory infinity is always actual infinity, i.e., completed infinity or esoteric infinity or finished infinity. > > What is WM's definition of a set being finite anyway. He has often been > asked for it, but I do not recall his ever giving it. See above. Regards, WM
From: mueckenh on 6 Feb 2007 05:18 On 4 Feb., 20:33, "William Hughes" <wpihug...(a)hotmail.com> wrote: > On Feb 4, 1:01 pm, mueck...(a)rz.fh-augsburg.de wrote: > > > On 4 Feb., 15:31, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > Similarly one can prove by induction: Every set of even positive > > > > numbers contains numbers which are larger than the cardinal number of > > > > the set. > > > > No. You make your usual mistake. > > > No, you make your usual mistake by thinking that a potentially > > infinite set can have an infinite cardinal number. > > I make no assumptions whatsoever about the cardinal > number of a potentially infinite set. (I do not even assume > that one exists). You said: "E contains numbers which are larger than the *cardinal number of E*." See below. > > > Let E be the potentailly infinite set of even numbers. > > > Two statements > > > > i: For every element e of E, the set > > > F(e) = {2,4,6,...,e} contains cardinal numbers which are > > > larger > > > than the cardinal number of F(e) > > > > ii E contains numbers which are larger than the cardinal > > > number of E > > > > Statement i is true and can be shown by induction. Statement > > > ii is false. > > > Of course, > > My claim is that statment ii is false. Your reply > is "of course". We both agree that statement ii > is false. Yes, because there is no cardinal number of E. > > > namely because there is no cardinal number for potentially > > infinite sets. > > > > The fact that statment i is true does not mean > > > that statement ii is true. > > > The fact that statement I is true means that there are no elements of > > the set which could increase its "number of elements" without > > increasing the sizes of elements in the set. > > As statement ii does not say or imply > anything about increasing the "number of elements" in E, > totally irrelevent. Statement ii is false. every statement beginning with "As statement ii does " therefore is irrelevant. > There is no element that can be shown to > be in E that is not in some F(e), but you need more than one > F(e) . There is no single F(e) that contains every element > that can be shown to be in E. Wong. There is no element in E which is outside of every F(e). Hence there are two possibilities: Either all elements of E exist and there is necessarily one F(E) or not all elements of E exist, then here is no F(E). > > The fact that > there is no element in E that is not in some set > F(e), does not mean that E has the same properties > as the sets F(e). > > The fact that statement i is true > does not imply that statement ii is true. > Statement i is true, statement ii is false > (note your "of course" above). > The statement "Every set of even positive > numbers contains numbers which are larger than the cardinal number of > the set", is false. Potential infinity is in this case the sequence of sets F(e) - there is no further set E other than this sequence. > The counterexample is the > potentially infinite set E. You seem to be unable to understand what potential infinity means. Therefore we should stop here. Regards, WM Regards, WM
From: Franziska Neugebauer on 6 Feb 2007 05:22 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer wrote: [...] >> > The simplest reason is that omega - n = omega for all n in N. >> >> Where did you get that from? Reference? EB? > > You could even figure it out by yourself. I cannot find any reference. Perhaps, there is none. F. N. -- xyz
From: mueckenh on 6 Feb 2007 05:28
On 5 Feb., 04:33, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <JCx39B....(a)cwi.nl> "Dik T. Winter" <Dik.Win...(a)cwi.nl> writes: > > In article <1170470698.824513.309...(a)m58g2000cwm.googlegroups.com> cbr...(a)cbrownsystems.com writes: > > > On Feb 2, 5:12 pm, Carsten Schultz <cars...(a)codimi.de> wrote: > > ... > > > > From that it follows that there are exactly four sets with four > > > > elements, since these are the elements of 4. It also follows that there > > > > is only one set with four elements, namely four. So 4=1. You should > > > > write a book about this. > > > > > > Dik is even reviewing it, I think. > > > > I am. I am now halfway chapter 8 and only found one serious error (in > > chapter 7). And one place where I have serious doubts (in chapter 8, but > > I have to look thoroughly at that). But until this point it is an > > excellent review about the history about the thinking about the infinite. > > Having read chapter 8 now completely, there are two errors there. Please let me know, and also that one in chapter 7. I would like to correct these errors in the second edition. > But > it remains an excellent review about the history of thinking. I just > started chapter 9, and, well, I will not say anymore now. I need a few > days for the remaining chapters. Attention. Dangerous contents! Regards, WM |