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From: Dik T. Winter on 15 Dec 2009 21:44 In article <9904ae9d-de2c-4d5f-86da-be165e8e9d7e(a)p30g2000vbt.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 15 Dez., 13:32, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > It is nonsense to define a pink unicorn. > > > > That statement is nonsense. Let pu be a pink unicorn is a proper > > definition. > > However there is nothing that satisfies that definition. > > That is a matter of taste. So finally you agree, you do not like to give definitions for things that do not exist (in some sense). However, in mathematics it is quite common to give definitions for things that do not exist, one of the jobs is to either proof that such a thing exist (according to the rules of mathematics) or that it does not exist. Giving only the field axioms, assume that S, being a set {a, b, c, d, e, f} with operations '+' and '*' be a field. Show that S does not exist. > > > The set N does not exist as > > > the union of its finite initial segments. This is shown by the (not > > > existing) path 0.000... in the binary tree. > > > > As you have defined your tree as only containing finite paths it is > > trivial to conclude that the infinite path 0.000... does not exist in > > your tree. > > However, this does *not* show that N does not exist. > > If N does exist, then the tree containing all finite paths contains > also infinite paths. No, you *explicitly* stated that your tree does not include infinite paths. If your tree also does include infinite paths you should not have stated a priori that it did not. That is why, in mathematics, there is a huge difference between the tree constructed as the collection of finite paths and as the collection of nodes where a path is allowed to start at the root and finish anywhere or not at all. > > > Let {1} U {1, 2} U {1, 2, 3} U ... = {1, 2, 3, ...}. > > > What then is > > > {1} U {1, 2} U {1, 2, 3} U ... U {1, 2, 3, ...} ? > > > > Ambiguous as stated. What is "..." standing for? > > Look two lines above: > > Let {1} U {1, 2} U {1, 2, 3} U ... = {1, 2, 3, ...} > hence > {1} U {1, 2} U {1, 2, 3} U ... U {1, 2, 3, ...} = {1, 2, 3, ...} . You mean: > ({1} U {1, 2} U {1, 2, 3} U ...) U {1, 2, 3, ...} = {1, 2, 3, ...} . yes? > > Not for "go on in the > > same way until", because when you go on in the same way you will never > > get at {1, 2, 3, ...}. Note this remark. If "..." is put between two things that means (in mathematics): go on the same until you come at what follows. > > > > > If it is the same, then wie have a stop in transfinite counting. > > > > However, if we try to attach a meaning we find that it is {1, 2, 3, ...}. > > Correct. > {1} U {1, 2} U {1, 2, 3} U ... = {1} U {1, 2} U {1, 2, 3} U ... U {1, > 2, 3, ...} No, correct mathematical notation is: > {1} U {1, 2} U {1, 2, 3} U ... = ({1} U {1, 2} U {1, 2, 3} U ...) U {1, > 2, 3, ...} > There is no space in the binary tree to contain infinite paths in > addition to the sequeneces of all finite paths. > The sequence 0.0, 0.00, 0.000, ... when being completely constructed, > is already the path 0.000... > You cannot add it separately. A sequence of paths is not a path. > Therefore: When all finite paths have been constructed within aleph_0 > steps, then all paths have been constructed. Here, again, you err. You can not construct something in aleph_0 steps; you will never complete your construction. You *cannot* get at aleph_0 step by step. > This hold for every limit of every sequence of finite paths. A limit is not a step by step process. -- 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: Jesse F. Hughes on 15 Dec 2009 21:52 "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > > > > > 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. But there is a difference. The ring axioms define a ring, while the axioms of set theory define a universe of sets, not a single set. So, a set is not something that satisfies the axioms of ZF. Rather, it is an object in a structure that satisfies the axioms of ZF. -- "Britney thought the idea of a pre-nup was vile, because she is loved-up with Kevin and cannot envisage breaking up. However, [...] no one in Hollywood these days get married without brokering a deal. [...] She had a long chat with Kevin and he was cool about it."
From: K_h on 15 Dec 2009 22:00 "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. If a sequence of sets, A_n, cannot be expressed as {X_n}, for some sequence of sets X_n, then limsup{n-->oo}A_n, liminf{n-->oo}A_n, and lim(n-->oo)A_n are defined by a specified wikipedia limit. Otherwise, let X_s, X_i, and L be the specified supremum, infimum, and limit (defined by wikipedia) on X_n. Define limsup{n-->oo}A_n = {X_s} liminf{n-->oo}A_n = {X_i} If L exists then lim(n-->oo)A_n = {L} otherwise it does not exist. k
From: Dik T. Winter on 15 Dec 2009 21:53 In article <Kup2uH.JC7(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > In article <r_6dnUXDv4pMfbvWnZ2dnUVZ_tWdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > > news:KunF0v.9y9(a)cwi.nl... > ... > > > By what definition is: > > > lim(n -> oo) n = N? > > > > Any of the wikipedia definitions. Here is the proof again. > ... > > 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? 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} = {} -- 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: K_h on 15 Dec 2009 22:02
"Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message news:Kup2uH.JC7(a)cwi.nl... > In article <r_6dnUXDv4pMfbvWnZ2dnUVZ_tWdnZ2d(a)giganews.com> > "K_h" <KHolmes(a)SX729.com> writes: > > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > > news:KunF0v.9y9(a)cwi.nl... > ... > > > By what definition is: > > > lim(n -> oo) n = N? > > > > Any of the wikipedia definitions. Here is the proof > > again. > ... > > 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. k |