Another AC anomaly?
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From: WM on 1 Dec 2009 07:42 On 1 Dez., 13:10, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <887fb198-aa2f-46ae-ab6a-91a67cb73...(a)u20g2000vbq.googlegroups..com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 30 Nov., 14:39, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > I think you are confusing the limit of a sequence of sets (which is a > > > set) and the limit of the sequence of the cardinalities of sets ( which > > > is a cardinality). =A0In general: the limit of the cardinalities is not > > > necessarily the cardinality of the limit, however much you would like > > > that to be the case. > > > > If the limit of cardinalities is 1, then the limit set has 1 element.. > > No because the limit of cardinalities is not necessarily the cardinality > of the limit, as I wrote just above. You may write this as often as you like, but you are wrong. If there is a limit set then there is a limit cardinality, namely the number of elements in that limit set. Everything else is nonsense. Regards, WM
From: Dik T. Winter on 1 Dec 2009 08:57 In article <6a0cfabf-c90e-4561-a45f-a4ffd33ca8e9(a)m3g2000yqf.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 1 Dez., 13:10, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <887fb198-aa2f-46ae-ab6a-91a67cb73...(a)u20g2000vbq.googlegroups= > .com> WM <mueck...(a)rz.fh-augsburg.de> writes: .... > > > If the limit of cardinalities is 1, then the limit set has 1 element. > > > > No because the limit of cardinalities is not necessarily the cardinality > > of the limit, as I wrote just above. > > You may write this as often as you like, but you are wrong. If there > is a limit set then there is a limit cardinality, namely the number of > elements in that limit set. Everything else is nonsense. Can you prove the assertion that the limit of the cardinalities is the cardinality of the limit? 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 = {} and so 2 = lim{n -> oo} | S(n) | != | lim{n -> oo} S_n | = 0 or can you show what I wrote is wrong? If so, what line is wrong? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: WM on 1 Dec 2009 12:04 On 1 Dez., 14:57, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <6a0cfabf-c90e-4561-a45f-a4ffd33ca...(a)m3g2000yqf.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 1 Dez., 13:10, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <887fb198-aa2f-46ae-ab6a-91a67cb73...(a)u20g2000vbq.googlegroups= > > .com> WM <mueck...(a)rz.fh-augsburg.de> writes: > ... > > > > If the limit of cardinalities is 1, then the limit set has 1 element. > > > > > > No because the limit of cardinalities is not necessarily the cardinality > > > of the limit, as I wrote just above. > > > > You may write this as often as you like, but you are wrong. If there > > is a limit set then there is a limit cardinality, namely the number of > > elements in that limit set. Everything else is nonsense. > > 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. Limit cardinality = Cardinality of the limit-set. If it does not exist, set theory claiming the existence of actual infinity is wrong. > 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). > If so, what line is wrong? Wrong is your definition of limit set, or set theory claiming actual infinity as substantially existing, or both. Regards, WM
From: William Hughes on 1 Dec 2009 12:12 On Dec 1, 1:04 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > Wrong WM uses the Wolkenmuekenheim definition of wrong Wrong = leads to a result that WM does not like. Inside Wolkenmuekenheim a result that WM does not like is a contradiction. Outside Wolkenmuekenheim there is a result that WM does not like but there is no contradiction. - William Hughes
From: George Greene on 1 Dec 2009 14:30
> 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. > 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. |