An uncountable countable set
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From: Virgil on 17 Oct 2006 01:19 In article <45343a7a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> No, set theory confuses the issue with its concentration on omega. > > > > Oh boy, here we go again with "Set theory confuses...". Please just way > > which axioms of set theory you reject and which ones you use instead. > > > > MoeBlee > > > > Uh, what axioms of set theory are specifically involved in your "proof"? > I don't remember a deduction from those axioms. Perhaps you could > refresh my memory. AS TO's memory is so full of holes, perhaps he should have it mended first. Provided it is not past mending.
From: Virgil on 17 Oct 2006 01:25 In article <45343aed(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > P.S. > > > > I lost the context, but somewhere you (Orlow) posted: > > > > "ZF and NBG don't handle sequences or their sums, but only unordered > > sets" > > > > Z set theory defines and proves theorems about ordered tuples, finite > > and infinite sequences, and infinite summations and infinite products > > and many other things like that. > > > > MoeBlee > > > > Huh! But I thought sets were unordered. TO's versions of set theory all required ordered sets, so why is he now objecting to ordered sets? And certainly N is ordered and well ordered, as are infinite sequences, series and products. > If the theory of infinite series > is derived from set theory, how come they seem to contradict each other > here? They do not seem to for anyone who understands them. For incompetents like TO, all sorts of perfectly natural and logical things may seem to be what they are not. > I don't recall a derivation or proof of the empty vase from the > axioms of set theory. TO has a lot of practice at forgetting important derivations and proofs.
From: Virgil on 17 Oct 2006 01:26 In article <45343ba4$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > MoeBlee wrote: > >> Tony Orlow wrote: > >>> No, set theory confuses the issue with its concentration on omega. > >> Oh boy, here we go again with "Set theory confuses...". Please just way > >> which axioms of set theory you reject and which ones you use instead. > > > > Good suggestion. But, what do you think the chance is that Tony will > > actually do that? > > > > I have done it lots of times. Never in any logical or coherent way.
From: Virgil on 17 Oct 2006 01:35 In article <45343c3d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> No, set theory confuses the issue with its concentration on omega. > >>> Oh boy, here we go again with "Set theory confuses...". Please just way > >>> which axioms of set theory you reject and which ones you use instead. > > > >> Good suggestion. But, what do you think the chance is that Tony will > >> actually do that? > > > > Tony has said that he does not like the axiom of infinity and > > the axiom of choice. Unlike many people who object to the > > axiom of infinity, Tony believes in infinite sets. He has more > > infinities than most. As for the axiom of choice, does > > the analysis of the balls in the vase problem use the axiom > > of choice? > > > > Stephen > > Thanks, Stephen. I am wondering the same thing. How does one formally > prove this from the axioms of ZFC? I believe this uses AoC, and no, I > don't like it. TO as usual; believes what he wants to believe despite the clear evidence of it being wrong. One is given in the original problem everything one needs to answer the original question, and nothing in either requires that one be able to choose an arbitrary element of any set. The only sets involved are subsets of the well ordered set of naturals, for which no extra choice functions to find the smallest member are ever necessary. > To me, what the Axiom of Choice really should be > conveying is a dimensional aspect to set Which is garbage. > which I think is something > it's used for, Which is garbage piled on garbage. > but I think it is abused beyond that. So, I have the same > question. How is this proven from the axioms? It has been proven here often enough so that if TO's memory were not as full of leaks as a sponge, he would by now be able to remember it.
From: Virgil on 17 Oct 2006 02:22
In article <45343dc9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > stephen(a)nomail.com wrote: > >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >>> MoeBlee wrote: > >>>> Tony Orlow wrote: > >>>>> No, set theory confuses the issue with its concentration on omega. > >>>> Oh boy, here we go again with "Set theory confuses...". Please just way > >>>> which axioms of set theory you reject and which ones you use instead. > >>> Good suggestion. But, what do you think the chance is that Tony will > >>> actually do that? > >> Tony has said that he does not like the axiom of infinity and > >> the axiom of choice. Unlike many people who object to the > >> axiom of infinity, Tony believes in infinite sets. He has more > >> infinities than most. As for the axiom of choice, does > >> the analysis of the balls in the vase problem use the axiom > >> of choice? > > > > My statement of the problem is > > > > Problem: For n = 1,2,..., define > > > > A_n = 12 - 1 / 2^(floor((n-1)/10)), > > R_n = 12 - 1 / 2^(n-1). > > > > For n = 1,2,..., define a function B_n by > > > > B_n(t) = 1 if A_n < t < R_n, > > 0 if t < A_n or t > R_n, > > undefined if t = A_n or t = R_n. > > > > Let V(t) = sum{n=1}^infty B_n(t). What is V(12)? > > > > Determining the value of V(12) certainly doesn't need the axiom of > > choice. > > > > What axioms ARE you using, specifically? None beyond ZF are needed for this. |