Cantor Confusion
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From: mueckenh on 11 Dec 2006 01:49 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > [...] > >> Again you have provided neither a definition of "number", nor of > >> "grow". > >> Are you unable to do so? In common parlance, but that is not > >> mathematics. In mathematics functions can grow in relation to their > >> argument, but not the entities they denote. > > > > Functions cannot grow, according to modern mathematics. > > Wrong. I have provided a definition: > > ,----[ <45742128$0$97220$892e7fe2(a)authen.yellow.readfreenews.net> ] > | Definition: A function f: A |-> B grows iff there exist a1 < a2 of > | dom(f) and f(a1) < f(a2). We use the abbreviation "f grows" for of > | "the function f grows". > `---- > The natural number n is a particular set of n elements. If the number n can take a value n_1 and can take a value n_2 with n_1 =/= n_2 then the number n can vary. > > The expression > > "variable" is merely a relict from ancient times when people knew that > > the objects of mathematics do not exist in some nirvana but have to be > > present in a mind where not everything can be present simultaneously. > > How do you call "Textbaustein" in English? Sorry, I did not expect that you read every word of mine addressed to other people. Regards, WM
From: mueckenh on 11 Dec 2006 01:50 William Hughes schrieb: > > > The reversal given was > > > > > > For every natural number n there exists a line L(n), such that > > > every natural number m <= n is an element of L(n) > > > > > > There exists a line L, such that for every natural number n, > > > every natural number m<=n, is contained in L. > > > > > > Note the movement of the phrase "every natural number". > > > > > > Please provide an alternate formulation that does not > > > involve the set of natural numbers. > > > > The set N can be involved and the quatifier can be changed as noted > > above as long as it is asured that: > > 1) every line has a finite number of elements > > 2) there is no element of the diagonal outside of every line. > > > > If you disagree please provide a counter example (with a finite line). > > The problem is not making a statment for each > finite line. The problem is combining all these statments > (one for each natural number n) into a single statement. Why should there be a problem as long as *all* lines are finite? Do you have a counter example where quantifier reversal in any finite line / set was prohibited? Regards, WM
From: Virgil on 11 Dec 2006 04:07 In article <1165819843.239428.298780(a)l12g2000cwl.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > The reversal given was > > > > > > > > For every natural number n there exists a line L(n), such that > > > > every natural number m <= n is an element of L(n) > > > > > > > > There exists a line L, such that for every natural number n, > > > > every natural number m<=n, is contained in L. > > > > > > > > Note the movement of the phrase "every natural number". > > > > > > > > Please provide an alternate formulation that does not > > > > involve the set of natural numbers. > > > > > > The set N can be involved and the quatifier can be changed as noted > > > above as long as it is asured that: > > > 1) every line has a finite number of elements > > > 2) there is no element of the diagonal outside of every line. > > > > > > If you disagree please provide a counter example (with a finite line). > > > > The problem is not making a statment for each > > finite line. The problem is combining all these statments > > (one for each natural number n) into a single statement. > > > Why should there be a problem as long as *all* lines are finite? Because what is being quantified is not line lenght, but line number, of which there are infinitely many. > Do you have a counter example where quantifier reversal in any finite > line / set was prohibited? There are lots of them for infinite sets, which is what is involved in this case. The statement is about every line, of which there are infinitely many, not every position in any one line. It is the case that for every line there are elements of the diagonal not in that line.
From: Han de Bruijn on 11 Dec 2006 04:52 stephen(a)nomail.com wrote: > Han.deBruijn(a)dto.tudelft.nl wrote: > >>step...(a)nomail.com schreef: > >>>Do you or do you not wish to abolish any mathematics >>>that involves infinity? If you are perfectly content >>>to let others freely explore whatever they wish, then >>>why are you so aggressive? > >>_What_ infinity. That's the question. Mainstream mathematics has mixed >>up infinity so much that it's not a sensible notion anymore. > > What is mixed up about infinity? Care to cite an example > where mainstream mathematics has mixed up infinity? See the many threads in this group, e.g. this one. > And > shouldn't people be free to explore whatever infinities > they wish? You were the one complaining that about > your freedoms being restricted, yet you seem perfectly > content to deny those freedoms to others. People should be free to explore whatever nonsense they wish, but please: don't call it mathematics. Han de Bruijn
From: Han de Bruijn on 11 Dec 2006 05:04
Virgil wrote: > In article <457c2070(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >>Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >>>David Marcus schreef: >>> >>>>Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>> >>>>>Ah, now don't act as if you didn't have those heated debates with some >>>>>manifest opponents of set theory, i.e. Wolfgang Mueckenheim. > > >>>>If you've been reading the threads, then you should know that WM has so >>>>far failed to present any mathematics at all. All he does is present >>>>incorrect arguments that he insists follow from the standard axioms and >>>>then proclaim: "Behold, standard mathematics is inconsistent." Big deal. >>> >>>Nonsense. The arguments presented by WM are quite reasonable. >> >>I agree. > > Now why does that fail to convince? It does fail to convince _you_, but it doesn't fail to convince some other people here. Han de Bruijn |