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From: Dik T. Winter on 16 Dec 2009 07:38 In article <k6idnaWIQNah0LXWnZ2dnUVZ_hqdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:Kup2G0.IBK(a)cwi.nl... > > In article <iNWdnfmPh7NpQ7vWnZ2dnUVZ_hydnZ2d(a)giganews.com> > > "K_h" <KHolmes(a)SX729.com> writes: > > > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > > > news:KunEFz.913(a)cwi.nl... > > ... > > > > The definition you provided for a sequence of sets A_n > > depends on whether > > each A_n is or is not a set containing a single set as an > > element. > > > > Your definition leads to some strange consequences. I can > > state the > > following theorem: > > > > Let A_n and B_n be two sequences of sets. Let A_s = lim > > sup A_n and > > A_i = lim inf A_n, similar for B_s and B_i. Let C_n be > > the sequence > > defined as: > > C_2n = A_n > > C_(2n+1) = B_n > > Theorem: > > lim sup C_n = union (A_s, B_s) > > lim inf C_n = intersect (A_i, B_i) > > Proof: > > easy. > > Yes, my definition did not include a limsup and liminf but > they can be added. With this addition, the limit of sets > like {X_n} is more in line with the general notion of a > limit. Well, the above theorem is still not valid with your definition. -- 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 16 Dec 2009 07:44 In article <hN-dneOj6K8oz7XWnZ2dnUVZ_rKdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:Kuq5DH.18H(a)cwi.nl... .... > > > This presupposes a particular construction for the > > > natural number. There are > > > other constructions that are consistent with ZF. Is the > > > limit valid for all > > > those possible models? > > > > For starters, try it with > > 0 = {} > > n+1 = {n} > > which is a valid construction of the naturals in ZF. > > > > Even with your definition > > lim sup(n -> oo) {n} = {} > > Why is it so important to you to have a limit definition and > a construction of the naturals such that lim(n->oo){n}={}? That is not important for me. With *your* definition and the construction of the natural numbers with n+1 = {n}, we *get* that lim(n -> oo) {n} = {} (note: set limit). > The general idea of a limit is that the limiting state is > what you get when you go through all sequences. If one > defines the naturals as you have done above then the general > notion of a limit suggests that the limiting state should be > something like: > > {...{{{{{{...{}...}}}}}}...} = limit > > We could construct a defintion of a limit so that this is > the end result but it may be that a better definition for > the limiting case of 0={} and n+1={n}is a defintion where > lim(n -> oo)n does not exist. We are talking about lim(n -> oo) {n} which is the limit of a sequence of sets, and not about lim(n -> oo) n which may or may not be the limit of a sequence of sets, depending on the actual construction of the natural numbers. But if I understand you well, your opinion is not that lim(n -> oo) {n} can be {N} or non-existing, depending on the way the natural numbers are constructed? -- 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 16 Dec 2009 07:40 In article <FbWdnevBWO8v0LXWnZ2dnUVZ_r-dnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:Kup2uH.JC7(a)cwi.nl... .... > > > Theorem: > > > lim(n ->oo) n = N. Consider the naturals: > > > > > > S_0 = 0 = {} > > > S_1 = 1 = {0} > > ... > > This presupposes a particular construction for the natural > > number. There are > > other constructions that are consistent with ZF. Is the > > limit valid for all > > those possible models? > > Why do you ask? There are many ways a limit can be defined > in ZF but the definition should embody the general idea of > what a limit is. But the definition ought to be such that the limit of a sequence does not depend on the exact construction of the sequence. That is that lim(n -> oo) {n} should be independent on the way the natural numbers are constructed. -- 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 16 Dec 2009 08:06 On 16 Dez., 03:15, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <364fe723-5ae4-4ed5-9219-c6b3892b3...(a)d21g2000yqn.googlegroups..com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 11 Dez., 03:40, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > > Without the axiom of infinity omega would not be immediately > > > > > existinging. > > > > > So apparently there is a definition of omega without the axiom of > > > > > infinity. > > > > > Can you state that definition? > > > > > > > > Look into Cantor's papers. Look into my book. > > > > > > I have never seen there a proper definition of omega. > Yes, so the definition uses the axiom of infinity. Without that axiom omega > would not be immediately existing, as I stated. No, you stated that you had never seen there a proper definition of omega. > But you thought there was > a definition without that axiom. I still have not found such a definition. > (The definition there is: > omega = {0, 1, 2, 3, ...} > and without the axiom of infinity it is not clear whether the right hand side > is a set.) > > > > > > There are no concepts of mathematics without definitions. > > > > > > > > So? What is a set? > > > > > > Something that satisfies the axioms of ZF for instance. > > > > Is that a definition? > > That is not something unheard of. In mathematics a ring is something that > satisfies the ring axioms, and that is pretty standard. And omega is something that does never end. > > > But in case you shouldn't have been able to find a definition of > > actual infinity, here is more than that: omega + 1. > > Ok, so actual infinity now is omega + 1. No, that's more than that. Omega already is actual infinity. And here are some other statements about actual infinity: Edward Nelson: Let us distinguish between the genetic, in the dictionary sense of pertaining to origins, and the formal. Numerals (terms containing only the unary function symbol S and the constant 0) are genetic; they are formed by human activity. All of mathematical activity is genetic, though the subject matter is formal. Numerals constitute a potential infinity. Given any numeraal, we can construct a new numeral by prefixing it with S. Now imagine this potential infinity to be completed. Imagine the inexhaustible process of constructing numerals somehow to have been finished, and call the result the set of all numbers, denoted by |N. Thus |N is thought to be an actual infinity or a completed infinity. This is curious terminology, since the etymology of infinite is not finished. We were warned. Aristotle: Infinity is always potential, never actual. Gauss: I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. We ignored the warnings. With the work of Dedekind, Peano, and Cantor above all, completed infinity was accepted into mainstream mathematics. Mathematics became a faith-based initiative. Try to imagine |N as if it were real. [...] [Edward Nelson (Department of Mathematics Princeton University): "Hilberts Mistake", Talk given at the Second New York Graduate Student Logic Conference, March 18, 2007.] Samul Feferman: "I am convinced that the platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject [...] platonism is the medieval metaphysics of mathematics; surely we can do better" [Solomon Feferman: "Infinity in Mathematics: Is Cantor Necessary?"] "The actual infinite is not required for the mathematics of the physical world" [Solomon Feferman: IN THE LIGHT OF LOGIC, p. 30] Regards, WM
From: Dik T. Winter on 16 Dec 2009 08:21
In article <a586deed-42c7-4523-acb2-1567183f04e1(a)g12g2000vbl.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 16 Dez., 03:15, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > > > > Without the axiom of infinity omega would not be immediatelry > > > > > > existinging. > > > > > > So apparently there is a definition of omega without the axiom > > > > > > of infinity. > > > > > > Can you state that definition? > > > > > > > > > > Look into Cantor's papers. Look into my book. > > > > > > > > I have never seen there a proper definition of omega. > > > Yes, so the definition uses the axiom of infinity. Without that axiom > > omega would not be immediately existing, as I stated. > > No, you stated that you had never seen there a proper definition of > omega. Look at the context. It started with my asking for a definition of omega without the axiom of infinity. You state that there is such a definition in your book. I find such a definition nowhere in your book. And the 'proper' should be read in the context of 'without the axiom of infinity'. > > > > > > There are no concepts of mathematics without definitions. > > > > > > > > > > So? What is a set? > > > > > > > > Something that satisfies the axioms of ZF for instance. > > > > > > Is that a definition? > > > > That is not something unheard of. In mathematics a ring is something that > > satisfies the ring axioms, and that is pretty standard. > > And omega is something that does never end. Whatever that may mean. > > > But in case you shouldn't have been able to find a definition of > > > actual infinity, here is more than that: omega + 1. > > > > Ok, so actual infinity now is omega + 1. > > No, that's more than that. Omega already is actual infinity. And here > are some other statements about actual infinity: So actual infinity is omega? > Let us distinguish between the genetic, in the dictionary sense of > pertaining to origins, and the formal. Numerals (terms containing > only > the unary function symbol S and the constant 0) are genetic; they are > formed by human activity. All of mathematical activity is genetic, > though the subject matter is formal. > Numerals constitute a potential infinity. Given any numeraal, we can > construct a new numeral by prefixing it with S. > Now imagine this potential infinity to be completed. Imagine the > inexhaustible process of constructing numerals somehow to have been > finished, and call the result the set of all numbers, denoted by |N. But N is *not* defined as an inexhaustable process of constructing numerals that somehow has been finished. The axiom of infinity state (together with the actual definition) state that it does exist, not how it is created. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |