Cantor Confusion
in [Math]
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From: mueckenh on 20 Feb 2007 09:05 On 19 Feb., 15:06, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1171890209.831371.70...(a)h3g2000cwc.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 19 Feb., 01:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > Not in my opinion, and I think not in mathematics. If the meaning of > > > 3 is "three of something" how than do we calculate "three of something" > > > times "three of something"? Or "nine of something" divided by > > > "three of something"? Or more concrete, if I divide nine apples by > > > three apples what is the result? > > > > three of something, where the "something" here means the unit. > > What unit? I One of something. Regards, WM
From: mueckenh on 20 Feb 2007 09:19 On 19 Feb., 15:19, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1171890516.474583.210...(a)p10g2000cwp.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 19 Feb., 01:31, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > > > > Can you tell me a form of set theory where 0 is *not* the first > > > > > > > ordinal or cardinal number? If so, how many elements does the > > > > > > > empty set have in such a system? > ... > > > > In all set theory 0 is called the first ordinal number, but in fact it > > > > is the zeroth one. Why do you start counting ordinals with 0 but start > > > > counting ordinally with 1? > > > > > > So your "natural number" above was a red herring? A human being in its > > > first year has the age 0. > > > > Before completing his first year, the being has the number of years > > which comes before 1. > > The first number drawn in lottery may be the 7. > > > > The first is that ordinal number which we start with. > > So your statement above: "In all set theory 0 is called the first ordinal > number, but in fact it is the zeroth one" was nonsense? No. The first is the first. The first is not the zeroest. It is nonsense to start counting by zero as is done in set theory. > > > > Indeed. There is only a definition of set. > > > > Not even that. > > <http://en.wikipedia.org/wiki/Set> LOL. Now, good old Cantor is good enough? He objected to numbers which are not finitely definable. Every user of his definition must agree with him because uncomputable reals are not "distinct objects of our perception or of our thought". By what means can two uncomputable numbers be distinguished? No, Dik. Either there is no definition of a set, then set theory may continue to exist as a matter of dreamwork. Or there is a firm definition of a set, then set theory crashes. Regards, WM
From: mueckenh on 20 Feb 2007 09:24 On 19 Feb., 15:23, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1171890647.125846.84...(a)h3g2000cwc.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 19 Feb., 01:35, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > The limit {1,2,3,...} = N does exist according to set theory. Why > > > > should the limit {1}, {1,2}, {1,2,3}, ... = N not exist? > > > > > > There you go again. Set theory does *not* use limits. I know of *no* > > > definition that gives the limit of a sequence of sets. So pray write > > > down here how *you* define the limit of a sequence of sets. > > > > We need not get involved into a discussion about limits of sets or the > > etymology of the limit ordinal number. > > That is irrelevant, you are using above undefined concepts. > > > Consider whether the following > > definitions of N in your opinion are correct or not. > > > > N = {n | n e N} = {0,1, 2, 3, ...} > > N = U {{0,1,2,...,n} | n e N} = U {{0}, {0,1}, {0,1,2}, ...} > > Both are correct. > > > If they are acceptable, then consider whether a set which contains all > > sets of the form {0,1,2,...,n} also contains N. > > As a subset (because every set is a subset of itself), not as an element. > > > And if every set which contains all sets of the form {0,1,2,...,n} > > contains N, > > As a subset. Fine. Every path of the tree is a special subset. (Not every subset is a path.) But there are only countably many finite subsets and countably many sets of subsets which belong to one and the same path. > > > why does the union of finite trees T(n) not contain an > > infinite path? > > I have never said that. I have stated that it *does* contain infinite > paths. So the union of finite trees U(T(n)) contains (as subsets) the path p(oo) and all its co-paths q(oo), ..., i.e., it contains P(oo)? Regards, WM
From: mueckenh on 20 Feb 2007 09:28 On 19 Feb., 17:46, Carsten Schultz <cars...(a)codimi.de> wrote: > mueck...(a)rz.fh-augsburg.de schrieb: > > > > > > > On 19 Feb., 01:35, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > >> In article <1171816407.391676.216...(a)v33g2000cwv.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > >> ... > >> > The limit {1,2,3,...} = N does exist according to set theory. Why > >> > should the limit {1}, {1,2}, {1,2,3}, ... = N not exist? > > >> There you go again. Set theory does *not* use limits. I know of *no* > >> definition that gives the limit of a sequence of sets. So pray write > >> down here how *you* define the limit of a sequence of sets. > > > We need not get involved into a discussion about limits of sets or the > > etymology of the limit ordinal number. Consider whether the following > > definitions of N in your opinion are correct or not. > > > N = {n | n e N} = {0,1, 2, 3, ...} > > N = U {{0,1,2,...,n} | n e N} = U {{0}, {0,1}, {0,1,2}, ...} > > All of them are true, of course. > > > If they are acceptable, then consider whether a set which contains all > > sets of the form {0,1,2,...,n} also contains N. > > And if every set which contains all sets of the form {0,1,2,...,n} > > contains N, why does the union of finite trees T(n) not contain an > > infinite path? > > Please clarify at which instances you mean `contains as subset' and at > which `contains as an element'. p(n) c T(n) p(oo) = U(p(n)) c U(T(n) = T(oo) Regards, WM
From: William Hughes on 20 Feb 2007 10:11
On Feb 20, 8:47 am, mueck...(a)rz.fh-augsburg.de wrote: > On 19 Feb., 14:36, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > Take a property X. Take a potentially infinite set > > A (say the union of all initial segments {1,2,3,...,n}). > > Then, as you note ("The claim in its generality is clearly wrong") > > the statements: > > > i: Every initial segment {1,2,3,...,n} has > > property X > > > ii: Every element of A that can be shown to exist > > is a natural number > > > Do not imply > > > iii: A has property X. > > > Sometimes i and ii are true and iii is true. > > Sometimes i and ii are true and iii is false. > > Statements i and ii cannot be used to prove iii. > > They can be used in certain cases. Not alone. i and ii are not enough to show iii. When someone says "iii is false" you reply "but i and ii are true". Since i and ii are not enough to show iii your replies are empty. > You wrote (then snipped) M: [t]he property that every set of even natural numbers must contain numbers M: larger than its cardinal number, is correct, unless the set contains M: unnatural numbers. As I noted this is false even in Wolkenmueckenheim. E is a counterexample. My question remains. How many times are you going to rephrase this and get it wrong? - William Hughes |