Another AC anomaly?
in [Logic]
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From: Jesse F. Hughes on 1 Dec 2009 14:40 George Greene <greeneg(a)email.unc.edu> writes: >> George Greene <gree...(a)email.unc.edu> writes: >> > Basically, any infinity of steps that has >> > a last element will have an answer to this question. >> > Any infinite sequence that does not have a last element >> > needs to get its "answer" from some NON-standard convention. > > On Nov 24, 2:30 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> No, that's not enough. >> >> Suppose that the light begins "on", at each step, I toggle the state. > > That was NOT the statement of the problem. > The statement of the problem was >>> you switch a light bulb on and off; > > This is inviting people to map to real-world binary light-switches, > NOT abstract unary togglers! If the light switch actually has > positions MARKED on and off (here they are usually up and down), > then this IS enough. > But of course you are right in principle that if "toggling the state" > AS OPPOSED to > "turning on and off" is what is going on, then, yes, we are still > confused. I nominate this post for "Most Spurious Attempt to Salvage Credibility" over the last 24 hour period. Since this is Usenet, that's a pretty high achievement. >> It seems that you agree that after omega-many steps, we do not know >> whether the light is on or off. But if we do not know at omega >> whether the light is on or off, then surely we do not know whether it >> is on or off at omega + 1. >> >> Right? > > Right, if the switch just toggles. > But if the switch actually has on and off positions and you are > just setting it to one, at every step, well, that's different. > Oddly. > I mean, it seems like it SHOULDN'T be different. > >> To put it differently, you claim "any infinity of steps that has >> a last element will have an answer to this question." w + 1 is an >> "infinity of steps" with a last element, but if we have an answer at >> w + 1, then we also have an answer at w. > > Well, if you're toggling, yes. > If you're turning on and off, then, well, it might still be possible > to turn > (e.g.) on, at w+1, even if you didn't know where you were at w. -- Jesse F. Hughes "Women aren't that unpredictable." "Well, I can't guess what you're getting at, honey." -- Hitchcock's _Rear Window_
From: Virgil on 1 Dec 2009 14:46 WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 1 Dez., 14:57, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > Can you prove the assertion that the limit of the cardinalities is the > > cardinality of the limit? > > With pleasure. My assertion is obvious if the limit set actually > exists. As there are two limits involved, it is not enough merely that the limit set exists. > Limit cardinality = Cardinality of the limit-set. That is what is to be proved: that the limit of the cardinalities equals the cardinality of the limit set. > If it does not exist, set theory claiming the existence of actual > infinity is wrong. Then in Wolkenmuekenheim it may be wrong, but that stll does not constitute a proof, nor anythng like a convincing argument. > > > I can prove that it can be false. As I wrote, > > given: > > S_n = {n, n+1} > > we have (by the definition of limit of sets: > > lim{n -> oo} S_n = {} > > This is wrong. You may find it helpful to see the approach where I > show a related example to my students. > > http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie 22 > > > and so > > 2 = lim{n -> oo} | S(n) | != | lim{n -> oo} S_n | = 0 > > or can you show what I wrote is wrong? > > Yes I can.* If actual infinity exists*, then the limit set exists. > Then the limit set has a cardinal number which can be determined > simply by counting its elements. > A simple example is > | lim[n --> oo] {1} | = | {1} | = lim[n --> oo] |{1}| = 1. > > Exchanging 1 by 2 or 3 or ... in the set {1} does not alter its > cardinality. > > If you were right, that | lim[n --> oo] S_n | =/= lim[n --> oo] |S_n| > was possible, then the limit set would not actually exist (such that > its elements could be counted and a different limit could be shown > wrong). What makes you think that the elements of a limit set can be counted? Outside of Wolkenmuekenheim that is false. > > > If so, what line is wrong? > > Wrong is your definition of limit set, or set theory claiming actual > infinity as substantially existing, or both. Or neither, at least outside of Wolkenmuekenheim
From: WM on 1 Dec 2009 16:26 On 1 Dez., 20:46, Virgil <Vir...(a)home.esc> wrote: > What makes you think that the elements of a limit set can be counted? If the set exists, its elements exist and can be counted. If it does not exist, then it has no cardinality and set theory is wrong. That fact has nothing at all to do with any quantifier magic used by some Fools Of Matheology. It is simply the question of to be or not to be (of sets and actual infinity). Regards, WM
From: Virgil on 1 Dec 2009 17:17 In article <f37ab501-a998-446b-8aaf-e88059d16a8d(a)z41g2000yqz.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 1 Dez., 20:46, Virgil <Vir...(a)home.esc> wrote: > > > What makes you think that the elements of a limit set can be counted? > > If the set exists, its elements exist and can be counted. That depends on what one means by "counting". Outside of Wolkenmuekenheim, there are sets which are not images of the naturals under any function, and such sets are uncountable.
From: Ross A. Finlayson on 1 Dec 2009 18:22
On Dec 1, 2:17 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <f37ab501-a998-446b-8aaf-e88059d16...(a)z41g2000yqz.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 1 Dez., 20:46, Virgil <Vir...(a)home.esc> wrote: > > > > What makes you think that the elements of a limit set can be counted? > > > If the set exists, its elements exist and can be counted. > > That depends on what one means by "counting". Outside of > Wolkenmuekenheim, there are sets which are not images of the naturals > under any function, and such sets are uncountable. No, then you would have proven ZFC consistent. Ross F. |