Another AC anomaly?
in [Logic]
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From: Virgil on 8 Dec 2009 20:29 In article <Hv2dnXQ7LtSxUIPWnZ2dnUVZ_hSdnZ2d(a)giganews.com>, "K_h" <KHolmes(a)SX729.com> wrote: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:KuAGqH.FrI(a)cwi.nl... > > In article <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dnZ2d(a)giganews.com> > > "K_h" <KHolmes(a)SX729.com> writes: > > ... > > > > > > When you mean with your statement about N: > > > > > > N = union{n is natural} {n} > > > > > > then that is not a limit. Check the definitions > > > > > > about > > > > > > it. > > > > > > > > > > It is a limit. That is independent from any > > > > > definition. > > > > > > > > It is not a limit. Nowhere in the definition of that > > > > union a limit is used > > > > or mentioned. > > > > > > Question. Isn't this simply a question of language? > > > > 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. > > He writes "...that is, lim a_n is obtained by taking the > union of all the sets a_n". The text book I have on set > theory defines N as the intersection of all inductive > subsets of any inductive set. So clearly there are many > equivalent approaches to defining N. In Rucker's approach, > we could define N as a limit: > > a_0 = {} //Zeroth member is the empty set. > > a_(n+1) = a_n Union {a_n} > > and then: > > N = limit a_n. > > My text book on set theory also explicitly states that we > can have a limit of a set of ordinals, for example: "...the > phase successor ordinal for an ordinal which is a successor > and limit ordinal for an ordinal which is a limit". In > fact, one of the problem sets is to prove the > bi-conditional: If X is a limit ordinal then UX=X (U is > union) and if UX=X then X is a limit ordinal. > > > > My > > > book on set theory defines omega, w, as follows: > > > > > > Define w to be the set N of natural numbers with its > > > usual order > > > < (given by membership in ZF). > > > > > > Now w is a limit ordinal so the ordered set N is, in the > > > ordinal sense, a limit. Of course w is not a member of > > > N > > > becasuse then N would be a member of itself (not allowed > > > by > > > foundation). > > > > Note here that N (the set of natural numbers) is *not* > > defined using a > > limit at all. That w is called a limit ordinal is a > > definition of the > > term "limit ordinal". It does not mean that the > > definition you use to > > define it actually uses a limit. (And if I remember > > right, a limit > > ordinal is an ordinal that has no predecessor, see, again > > no limit > > involved.) > > N can be defined as a limit or not as a limit. These are > really equivalent approaches. 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. Unless some other definition replaces this one, it is impossible to form the union of a family of sets unless that family are already the members of a set. In which case, N cannot be defined as the limit you suggest, as it would have to exist before it exists.
From: K_h on 9 Dec 2009 01:25 "Virgil" <Virgil(a)home.esc> wrote in message news:Virgil-2C0D8F.18292508122009(a)newsfarm.iad.highwinds-media.com... > In article > <Hv2dnXQ7LtSxUIPWnZ2dnUVZ_hSdnZ2d(a)giganews.com>, > "K_h" <KHolmes(a)SX729.com> wrote: > >> "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message >> news:KuAGqH.FrI(a)cwi.nl... >> > In article >> > <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dnZ2d(a)giganews.com> >> > "K_h" <KHolmes(a)SX729.com> writes: >> > ... >> > > > > > When you mean with your statement about N: >> > > > > > N = union{n is natural} {n} >> > > > > > then that is not a limit. Check the >> > > > > > definitions >> > > > > > about >> > > > > > it. >> > > > > >> > > > > It is a limit. That is independent from any >> > > > > definition. >> > > > >> > > > It is not a limit. Nowhere in the definition of >> > > > that >> > > > union a limit is used >> > > > or mentioned. >> > > >> > > Question. Isn't this simply a question of language? >> > >> > 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. >> >> He writes "...that is, lim a_n is obtained by taking the >> union of all the sets a_n". The text book I have on set >> theory defines N as the intersection of all inductive >> subsets of any inductive set. So clearly there are many >> equivalent approaches to defining N. In Rucker's >> approach, >> we could define N as a limit: >> >> a_0 = {} //Zeroth member is the empty set. >> >> a_(n+1) = a_n Union {a_n} >> >> and then: >> >> N = limit a_n. >> >> My text book on set theory also explicitly states that we >> can have a limit of a set of ordinals, for example: >> "...the >> phase successor ordinal for an ordinal which is a >> successor >> and limit ordinal for an ordinal which is a limit". In >> fact, one of the problem sets is to prove the >> bi-conditional: If X is a limit ordinal then UX=X (U is >> union) and if UX=X then X is a limit ordinal. >> >> > > >> > > My >> > > book on set theory defines omega, w, as follows: >> > > >> > > Define w to be the set N of natural numbers with >> > > its >> > > usual order >> > > < (given by membership in ZF). >> > > >> > > Now w is a limit ordinal so the ordered set N is, in >> > > the >> > > ordinal sense, a limit. Of course w is not a member >> > > of >> > > N >> > > becasuse then N would be a member of itself (not >> > > allowed >> > > by >> > > foundation). >> > >> > Note here that N (the set of natural numbers) is *not* >> > defined using a >> > limit at all. That w is called a limit ordinal is a >> > definition of the >> > term "limit ordinal". It does not mean that the >> > definition you use to >> > define it actually uses a limit. (And if I remember >> > right, a limit >> > ordinal is an ordinal that has no predecessor, see, >> > again >> > no limit >> > involved.) >> >> N can be defined as a limit or not as a limit. These are >> really equivalent approaches. > > 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 > Unless some other definition replaces this one, it is > impossible to form > the union of a family of sets unless that family are > already the > members of a set. > > In which case, N cannot be defined as the limit you > suggest, as it would > have to exist before it exists. Here is one way to define N as a limit. - Given a set x, the successor of x is the set x'=xU{x}. - A set y is inductive if x' is in y whenever x is in y. - Given an initial set x, the inductive set comprised of the successors of x is called a limit set of the sequence of sets x'=xU{x}, x''=x'U{x'}, ... . - 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 It should be pointed out that N is a limit set even if N is initially given by a definition that doesn't involve the notion of a limit. Here is another way to see that N is a limit even if you consider it bad taste to define it in those terms: Let \/ = union Let /\ = Intersection Define infimum and supremum as follows: liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m] (n-->oo) limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m] (n-->oo) If these two are the same then the limit exists and is both of them. So, let's consider the naturals: X_0 = 0 = {} X_1 = 1 = {0} X_2 = 2 = {0,1} X_3 = 3 = {0,1,2} X_4 = 4 = {0,1,2,3} X_5 = 5 = {0,1,2,3,4} .... Evaluate the liminf case: /\(m=0, oo) X_m = X_0 /\ X_1 /\ X_2 ... = {} /\ {0} /\ {0,1} /\ {0,1,2} /\ ... = {} /\(m=1, oo) X_m = X_1 /\ X_2 /\ X_3 ... = {0} /\ {0,1} /\ {0,1,2} /\ ... = {0} /\(m=2, oo) X_m = X_2 /\ X_3 /\ X_4 ... = {0,1} /\ {0,1,2} /\ {0,1,2,3} /\ ... = {0,1} .... etc. Now, we \/(n=0, oo) to union all these together and we get N: \/(n=0, oo) = {} \/ {0} \/ {0,1} \/ ... \/(n=0, oo) = {0,1,2,3,...} \/(n=0, oo) = N Next, evaluate the limsup case: \/(m=0, oo) X_m = X_0 \/ X_1 \/ X_2 ... = {} \/ {0} \/ {0,1} \/ {0,1,2} \/ ... = {0,1,2,3,...} = N \/(m=1, oo) X_m = X_1 \/ X_2 \/ X_3 ... = {0} \/ {0,1} \/ {0,1,2} \/ ... = {0,1,2,3,...} = N \/(m=2, oo) X_m = X_2 \/ X_3 \/ X_4 ... = {0,1} \/ {0,1,2} \/ {0,1,2,3} ... = {0,1,2,3,...} = N .... etc. Now, we /\(n=0, oo) intersect all of these to get: /\(n=0, oo) = N /\ N /\ N /\ ... /\(n=0, oo) = N Limsup and liminf both give N and so the limit is N. k
From: Virgil on 9 Dec 2009 02:35 In article <c8idnc6JG6hv34LWnZ2dnUVZ_q-dnZ2d(a)giganews.com>, "K_h" <KHolmes(a)SX729.com> wrote: > "Virgil" <Virgil(a)home.esc> wrote in message > news:Virgil-2C0D8F.18292508122009(a)newsfarm.iad.highwinds-media.com... > > In article > > <Hv2dnXQ7LtSxUIPWnZ2dnUVZ_hSdnZ2d(a)giganews.com>, > > "K_h" <KHolmes(a)SX729.com> wrote: > > > >> "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > >> news:KuAGqH.FrI(a)cwi.nl... > >> > In article > >> > <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dnZ2d(a)giganews.com> > >> > "K_h" <KHolmes(a)SX729.com> writes: > >> > ... > >> > > > > > When you mean with your statement about N: > >> > > > > > N = union{n is natural} {n} > >> > > > > > then that is not a limit. Check the > >> > > > > > definitions > >> > > > > > about > >> > > > > > it. > >> > > > > > >> > > > > It is a limit. That is independent from any > >> > > > > definition. > >> > > > > >> > > > It is not a limit. Nowhere in the definition of > >> > > > that > >> > > > union a limit is used > >> > > > or mentioned. > >> > > > >> > > Question. Isn't this simply a question of language? > >> > > >> > 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. > >> > >> He writes "...that is, lim a_n is obtained by taking the > >> union of all the sets a_n". The text book I have on set > >> theory defines N as the intersection of all inductive > >> subsets of any inductive set. So clearly there are many > >> equivalent approaches to defining N. In Rucker's > >> approach, > >> we could define N as a limit: > >> > >> a_0 = {} //Zeroth member is the empty set. > >> > >> a_(n+1) = a_n Union {a_n} > >> > >> and then: > >> > >> N = limit a_n. > >> > >> My text book on set theory also explicitly states that we > >> can have a limit of a set of ordinals, for example: > >> "...the > >> phase successor ordinal for an ordinal which is a > >> successor > >> and limit ordinal for an ordinal which is a limit". In > >> fact, one of the problem sets is to prove the > >> bi-conditional: If X is a limit ordinal then UX=X (U is > >> union) and if UX=X then X is a limit ordinal. > >> > >> > > > >> > > My > >> > > book on set theory defines omega, w, as follows: > >> > > > >> > > Define w to be the set N of natural numbers with > >> > > its > >> > > usual order > >> > > < (given by membership in ZF). > >> > > > >> > > Now w is a limit ordinal so the ordered set N is, in > >> > > the > >> > > ordinal sense, a limit. Of course w is not a member > >> > > of > >> > > N > >> > > becasuse then N would be a member of itself (not > >> > > allowed > >> > > by > >> > > foundation). > >> > > >> > Note here that N (the set of natural numbers) is *not* > >> > defined using a > >> > limit at all. That w is called a limit ordinal is a > >> > definition of the > >> > term "limit ordinal". It does not mean that the > >> > definition you use to > >> > define it actually uses a limit. (And if I remember > >> > right, a limit > >> > ordinal is an ordinal that has no predecessor, see, > >> > again > >> > no limit > >> > involved.) > >> > >> N can be defined as a limit or not as a limit. These are > >> really equivalent approaches. > > > > 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 > > > Unless some other definition replaces this one, it is > > impossible to form > > the union of a family of sets unless that family are > > already the > > members of a set. > > > > In which case, N cannot be defined as the limit you > > suggest, as it would > > have to exist before it exists. > > Here is one way to define N as a limit. > > - Given a set x, the successor of x is the set x'=xU{x}. > > - A set y is inductive if x' is in y whenever x is in y. > > - Given an initial set x, the inductive set comprised of the > successors of x is called a limit set of the sequence of > sets x'=xU{x}, x''=x'U{x'}, ... . It is the set of all of them, if that is what you mean by union > > - 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. > > It should be pointed out that N is a limit set even if N is > initially given by a definition that doesn't involve the > notion of a limit. 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. Here is another way to see that N is a > limit even if you consider it bad taste to define it in > those terms: > > Let \/ = union > > Let /\ = Intersection > > Define infimum and supremum as follows: > > liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m] > (n-->oo) > > limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m] > (n-->oo) > > If these two are the same then the limit exists and is both > of them. 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.
From: WM on 9 Dec 2009 09:27 On 9 Dez., 08:35, Virgil <Vir...(a)home.esc> wrote: > In article <c8idnc6JG6hv34LWnZ2dnUVZ_q-dn...(a)giganews.com>, > 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. 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 lim inf{n -> oo} S_n contains those elements that occur in all S_n from a certain S_n (which can be different for each element). lim{n -> oo} S_n exists whenever lim sup and lim inf are equal. > > > Define infimum and supremum as follows: > > > liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m] > > (n-->oo) > > > limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m] > > (n-->oo) > > > If these two are the same then the limit exists and is both > > of them. > > 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 If the limit set emerges continuously (i.e. growing one-by-one) from the sets of the sequence, then the cardinality will be bound to this process and will also continuously emerge from the cardinalities of the sets of the sequence. If there is a gap in one sequence and no gap in the other, then this indicates that at least one of the limits does not exist, probably both do not exist. This has been substanciated by the following thought-experiment: If N is generated as a limit, then it is generated by adding number after number. 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. Therefore neither N nor Card(N) are meaningful concepts - at least not in mathematics. Regards, WM
From: Virgil on 9 Dec 2009 15:34
In article <b6fe23bb-f753-4cfc-a9d6-79711f3a1e5d(a)r40g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 9 Dez., 08:35, Virgil <Vir...(a)home.esc> wrote: > > In article <c8idnc6JG6hv34LWnZ2dnUVZ_q-dn...(a)giganews.com>, > > > > 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. > > 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 > lim inf{n -> oo} S_n contains those elements that occur in all S_n > from > a certain S_n (which can be different for each > element). > lim{n -> oo} S_n exists whenever lim sup and lim inf are equal. > > > > > Define infimum and supremum as follows: > > > > > liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m] > > > (n-->oo) > > > > > limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m] > > > (n-->oo) > > > > > If these two are the same then the limit exists and is both > > > of them. > > > > 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 > > If the limit set emerges continuously (i.e. growing one-by-one) from > the sets of the sequence, then the cardinality will be bound to this > process and will also continuously emerge from the cardinalities of > the sets of the sequence. > > If there is a gap in one sequence and no gap in the other, then this > indicates that at least one of the limits does not exist, probably > both do not exist. > > This has been substanciated by the following thought-experiment: > If N is generated as a limit, then it is generated by adding number > after number. > 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. Since WM's supposed 'lessons' are only valid in Wolkenmuekenheim, if even there, that is irrelevant. > > Therefore neither N nor Card(N) are meaningful concepts - at least not > in mathematics. They are in most of the mathematics outside of Wolkenmuekenheim. WM's attempts to impose his personal worldview on the world again fails. The only place it has any chance is with those naive captive audiences of students in his classrooms. |