An uncountable countable set
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From: Virgil on 19 Aug 2006 14:30 In article <ec7gek$frg$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > You said that a set exists if all its elements do exist. If so, then > > all natural numbers do exist and make up an infinite number without one > > of them being infinite. > > Inductively provable: Any set of consecutive naturals starting at 1 > always has its largest element equal to size of the set. Equally provable: Any finitely bounded set of consecutive ordinals starting with the first always has a largest element less than its "size".
From: Virgil on 19 Aug 2006 14:34 In article <ec7hac$h84$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > It is only a fact given one's definition of the infinitude of a set. By > the accepted set theoretical system of transfinite cardinalities, sure, > you're correct. But it doesn't jibe with more traditional, more > tried-and-true quantitative modes of thought. Formulas go out the window > with transfinite set theory, elements are added without increasing the > set, and computation fails. Why else would there be such fervent > objections to Cantorian hocus pocus? See? Even avid infinitrons like > myself and devout finitists like WM see the same problem. Once we get > Cantor out of the way, then WM and I will have to duke it out, eh? > ;) Cantor's groundwork has survived attack by both more competent and less competent critics that either of you. So that Cantor will be remembered and honored long after both of you are long forgotten.
From: Virgil on 19 Aug 2006 14:37 In article <ec7hej$h84$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec7d7r$ath$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >> Virgil wrote: > >>> In article <ec5k1i$4gr$2(a)ruby.cit.cornell.edu>, > >>> Tony Orlow <aeo6(a)cornell.edu> wrote: > >>> > >>> > >>>> By the way, hello! Nice to see you all again, I think. > >>> > >>> Does Tony actually think? > >> Nice to see you too, Virgil. I don't think > > > > I thought not. > > Well, that was lame. You can do better. (I hope) If TO were ever to produce anything honestly mathematical, I might have something to work with. As long as TO sticks to producing nothing but garbage, it is GIGO.
From: Tony Orlow on 19 Aug 2006 15:30 Virgil wrote: > In article <ec7gek$frg$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > > >>> You said that a set exists if all its elements do exist. If so, then >>> all natural numbers do exist and make up an infinite number without one >>> of them being infinite. >> Inductively provable: Any set of consecutive naturals starting at 1 >> always has its largest element equal to size of the set. > > Equally provable: Any finitely bounded set of consecutive ordinals > starting with the first always has a largest element less than its > "size". And, your position is that, because the ordinals start with 0, rather than 1, the addition of that single element makes the set larger than any finite element contained therein? Then, you must be of the opinion that adding a single element can turn a finite set into an infinite one, which means there must be some largest finite natural which you can increment to produce a non-finite value. Pray tell, what be's this value, O Wise and Powerful Virgil? :) TO
From: Tony Orlow on 19 Aug 2006 15:33
Virgil wrote: > In article <1156000079.633380.83620(a)p79g2000cwp.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Dik T. Winter schrieb: >>> Oh. I must have missed something, because I have not seen a proof. >>> 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)} >> >> Regards, WM > > What does "M_f(n)" mean? As far as I can see, M_f or M(f) is set, not a > function so "M_f(n)" is not even defined, much less {f,n,M_f(n)). > > In any case, there are lots of injections, f, from N to P(N). > > Consider f(n) = {n}, for example. > In which case one easily sees that > M(f) = {}. > > Which exists quite nicely, thank you very much! 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. TO |