In article <e8d6cf87-4adf-492a-be49-04a4106e2dd5@z10g2000yqb.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 23 Jun., 14:57, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > > Tim Little <t...@little-possums.net> writes: > > > In fact it would be a lot easier if instead of reals, we talked about > > > sets of natural numbers, and instead of computable reals, we talked > > > about recursively enumerable sets of natural numbers. > > > > It would be even more easy if instead of the uncountability of the reals > > we talked about the standard proof of Cantor's theorem: > > > > Let f be a function taking elements of a set A to subsets of A. There > > is then a subset of A not in the range of f. For consider the set > > > > D = {x in A | x not in f(x)}. > > > > There is no a in A such that f(a) = D: if there were, we'd have a in > > f(a) iff a not in f(a), a contradiction. > > That could be a good proof, if we knew that all subsets of an infinite > set A would exist.
> But already Fraenkel wrote in the third edition of > his famous book (1928, p. 279f) with respect to the axiom of power > set, that: der Begriff "Teilmenge³ eine andere, wesentlich engere > Bedeutung hat als in der CANTORschen Mengenlehre. In dieser konnten > wir bei der Bildung der Potenzmenge Um eine beliebige Gesamtheit von > Elementen aus m zu einer Teilmenge von m zusammenfassen und waren dann > sicher, daß diese sich unter den Elementen von Um findet. Jetzt ist > uns eine derartige, weitgehende Freiheit gewährende "Bildung³ einer > Teilmenge von m nicht gestattet, also auch ihr Auftreten unter den > Elementen von Um keineswegs gesichert. (Contrary to the second edition > of 1923, Fraenkel now knew Skolem's proof of the same year and had to > explain how it could be circumvented.) > > Therefore, there is not every subset of an infinite set. Why then > should exist the subset of A that contains its pre-image if it does > not contain it, and does not contain ist, if it contains it?
Why should it not exist? My own take on set theory is that everything should be allowed that does not allow proofs of statements of the form "P and not P".
WM is a minimalist, but offers nothing but his own prejudices as to why he should be able to impose his views on everyone.
It is not as if there were anything like Russell's paradox that results from, say, FOL+ZFC. If there ever were, that might justify WM's paranoia re infiniteness, but it certainly hasn't happened yet.
