An uncountable countable set
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From: David R Tribble on 21 Aug 2006 15:55 Tony Orlow wrote: > But Monsieur, what about the injection from P(N) into N, via the bit > strings which denote set membership, each of which also corresponds to a > binary natural? Tsk, tsk. Mustn't forget that one! Remember, the only > set which doesn't map is the entire set, and that maps to the largest > natural, that is, ...1111 with all bits in finite positions. .... as well as all the infinite subsets of N. You keep forgetting about those, don't you?
From: Virgil on 21 Aug 2006 15:57 In article <1156149770.217482.166370(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > What is the sum of infinitely many finite numbers? For suitable infinite sets of real numbers the sum can be any real number one chooses.
From: Virgil on 21 Aug 2006 16:08 In article <1156149770.217482.166370(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1156000079.633380.83620(a)p79g2000cwp.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > Given an injection f: N -> P(N), why does the set > > > > M(f) = {n in N | n !in f(n)} > > > > not exist? > > > > > > Excuse me, the non-existing set is the triple {f, n, M_f(n)} > > > > Excuse me, Hessenberg was not talking about such triples. > > But I am doing so in order to show that his arguing concerns > impredicative definitions and is inconclusive.f is the mapping, n is a > natural number, M_f(n) is a set which contains all nongenerators, > including n if not including n which is mapped on M. For injections f:N --> P(N) which are not required to be surjections there is no difficulty with the existence of M(f). But "Mueckenh"'s 'M_f(n)' has no unambiguous meaning. Does "Mueckenh" mean that M_f is a function with M_f(n) as its value? If so, what are the domain and codomain of M_f, and how is the value M_f(n) supposed to be determined? Or does "Mueckenh" mean that f(n), as a member of P(N), is being used to determine some M_f(n)? If so what id M_f(n) ? a member of N? a member of P(N)? something else?
From: Virgil on 21 Aug 2006 16:10 In article <1156163893.101419.232240(a)p79g2000cwp.googlegroups.com>, "Albrecht" <albstorz(a)gmx.de> wrote: > Infinite sets are self contradicting. Not in ZF or NBG. What are the axioms of Storz's system?
From: mueckenh on 21 Aug 2006 16:27
Franziska Neugebauer schrieb: > > Consider a staircase: > > Set theory is not about staircases. It has absolutely no application? Not even in finite sets? What a pity! > > If you climb 10 meters, then you arrive at height 10 m. > > But you climb infinitely many meters without arriving at an infinite > > height? > > I don't discuss physicalisms. Set theory does not claim to rule physics > therefore "physical" arguments are unapt to explore set theory. O, I thought in finite cases set theory were true? You should know that Cantor created set theory for this sake... But, alas, you don't like him. > > > Fact is: if omega is a number, then omega differences of height 1 > > result in the total size omega, because > > > > omega * 1 = omega. > > > > Isn't it a fact? > > What are you talking about? How omega differences of 1 supply a number omega. But you don't discuss numbers, I suppose? Regards, WM ____________ To all: Sorry, the thread has become to vivid for me. I am no longer able to answer all contribution directed to me. i am in vacations. |