Another AC anomaly?
in [Logic]
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From: WM on 10 Dec 2009 08:19 On 10 Dez., 08:19, Virgil <Vir...(a)home.esc> wrote: > In article > <b963b9aa-c345-43cf-bf78-e9e27401f...(a)c34g2000yqn.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 10 Dez., 01:58, "K_h" <KHol...(a)SX729.com> wrote: > > > > > When using an intermediate reservoir, as shown in my > > > > lesson > > > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie > > > > 22 > > > > it becomes clear that N cannot be generated by adding > > > > number after > > > > number. > > > > Why not? Say we have an infinitely large sheet of paper and > > > we print each natural number, n, on the paper at time > > > t=1-1/(n+1). Certainly at time t=1 we have all the naturals > > > printed on the page. > > > It seems so. But it is wrong. You see it if you consider the > > alternative process using an intermediate reservoir as "realized" in > > my lesson above. > > > naturals - reservoir - paper > > N - { } - { } > > N/{1} - {1} - { } > > N/{1,2} - {2} - {1} > > N/{1,2,3} - {3} - {1,2} > > ... > > N/{1,2,3, ...,n} - {n} - {1,2,3, ...,n-1} > > ... > > > The set in the middle contains a number at every time after t = 0. > > Hence this number cannot yet have been printed on the paper (because > > it will be printed only after its follower will have entered the > > reservoir). > > And when all the numbers have passed through your "reservoir", both into > and out ofd it, as will have happened by t = 1, which numbers does WM > claim will still be unprinted. Then a last one would have had to leave the intermediate reservoir. > > When going through the terms of an infinite sequence, as in the above, > EITHER the process hangs up on a particular term of that sequence OR it > goes through every term of the sequence, TERTIUM NON DATUR.- Or it is proved that the assumption of finished infinity is nonsense. Secundum non datur. Regards, WM
From: Dik T. Winter on 10 Dec 2009 08:54 In article <Hv2dnXQ7LtSxUIPWnZ2dnUVZ_hSdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:KuAGqH.FrI(a)cwi.nl... .... > > Not at all. When you define N as an infinite union there > > is no limit > > involved, there is even no sequence involved. N follows > > immediately > > from the axioms. > > I disagree. Please note that I am not endorsing many of > WM's claims. There are many equivalent ways of defining N. > I have seen the definition that Rucker uses, in his infinity > and mind book, in a number of books on mathematics and set > theory: On page 240 of his book he defines: > > a_(n+1) = a_n Union {a_n} > > and then: > > a = limit a_n. But here an infinite union is *not* involved, that is the crucial difference. As stated, you may define N as a limit or not, and when it is defined as an infinite union as in: N = union {1, 2, ..., n} a limit is not involved. -- 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: Dik T. Winter on 10 Dec 2009 09:06 In article <Virgil-1E8B09.00355309122009(a)newsfarm.iad.highwinds-media.com> Virgil <Virgil(a)home.esc> writes: > In article <c8idnc6JG6hv34LWnZ2dnUVZ_q-dnZ2d(a)giganews.com>, > "K_h" <KHolmes(a)SX729.com> wrote: .... > > > In ZF, unions are defined only for sets of sets and for > > > such a set of > > > sets S, the union is defined a the set of all elements of > > > elements of S. > > > > Check out: > > http://planetmath.org/encyclopedia/SequenceOfSetsConvergence.html With that definition set inclusion is a requirement, with the definition I gave it is not. > > - Let N be the limit set formed from the initial set {}. > > > > In this case N is a convergent set: > > > > http://planetmath.org/encyclopedia/SequenceOfSetsConvergence.html > > However, in the discussion between Dik and WM, Dik gave SPECIFIC > definition of what HE meant by the limit of a seqeunce of sets which > differed from that in your citation. The difference is that with my definition (which is also quite generally used) sequences of sets can also converge if there is no set inclusion, as in the sequence {n}. If A_n subset A_(n+1) or A_(n+1) subset A_n the two definitions are equivalent. > The issue is not whether the naturals are such a limit but whether for > every so defined limit the cardinality of the limit equals the limit > cardinality of their cardinalities, which is different sort of limit. No, actually the issue is whether all possible definitions of N include a limit, and that is false: N is the smallest inductive set involves no limit at all. And WM's definitions of N did *not* include a limit. He thought so because he mistakenly thought that an infinite union does imply a limit. This from the mistaken thought that uniting a collection of sets goes stepwise. (And note, Virgil, WM uses the also common notation: union(i in I) {S_i} as shorthand for union{ S_i | i in I} where the latter is standard ZF. I.e. the uniting is about sets rather than about members of a single set. -- 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: Dik T. Winter on 10 Dec 2009 09:40 In article <QP-dnV0EIYPt2b3WnZ2dnUVZ_h2dnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > "Virgil" <Virgil(a)home.esc> wrote in message > news:Virgil-1E8B09.00355309122009(a)newsfarm.iad.highwinds-media.com... .... > > The issue between Dik and WM is whether the limit of a > > sequence of sets > > according to Dik's definition of such limits is > > necessarily the same as > > the limit of the sequence of cardinalities for those sets. > > > > And Dik quire successfully gave an example in which the > > limits differ. > > I suspect those definitions are not valid. The definition I > used is the one on wikipedia and is generally `standard' -- > as I've seen it in numerous places, including books and > websites. Have a look at <http://en.wikipedia.org/wiki/Lim_inf> in the section titled "Special case: dicrete metric". An example is given with the sequence {0}, {1}, {0}, {1}, ... where lim sup is {0, 1} and lim inf is {}. Moreover, in what way can a definition be invalid? -- 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: Dik T. Winter on 10 Dec 2009 09:45
In article <k_udnSOIvMxf2r3WnZ2dnUVZ_gqdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: .... > > Dik: > > Given a sequence of sets S_n then: > > lim sup{n -> oo} S_n contains those elements that occur > > in > > infinitely > > many S_n > > I don't think this is a good definition for the limit of a > sequence of sets S_n. That is not the definition of a limit but of lim sup. Consider the sequence of numbers 0, 1, 0, 1, ..., the lim sup of this sequence is 1, the lim inf is 0 and as those two are not equal, the limit does not exist. > For instance, consider the > alternating sets: > > S_0 = {0, 2, 4, 6, 8,...} > S_1 = {-1, -3, -5, ...} > S_2 = {0, 2, 4, 6, 8,...} > S_3 = {-1, -3, -5, ...} > S_4 = {0, 2, 4, 6, 8,...} > S_5 = {-1, -3, -5, ...} > ... The lim sup is the set {..., -5, -3, -1, 0, 2, 4, 6, ...}, the lim inf is {} and as these two are not equal the limit does not exist. > Like Thompson's lamp alternating between on and > off it reaches no limit in any intuitive sense of what a > limit is. But it has a lim sup and a lim inf. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |