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From: Virgil on 22 Oct 2005 02:41 In article <1129946611.902596.262670(a)g49g2000cwa.googlegroups.com>, "William Hughes" <wpihughes(a)hotmail.com> wrote: > Dave Rusin wrote: > > In article <1129901285.084347.34810(a)g49g2000cwa.googlegroups.com>, > > William Hughes <wpihughes(a)hotmail.com> wrote: > > > > >Yes all sets have a cardinality. > > > > That's your Choice. > > Am I assuming AC here? > > My understanding was that, while AC was needed to > put a total ordering on the equivalence classes > under bijection, one could assert the existence > of these classes even without AC. Or are the > cardinalities defined not as all possible > equivalence classes but as the equivalence classes > on the ordinals? > > -William Hughes As I understand it, without AC, one cannot be sure of creating any sort of function at all from one arbitrary set to another, much less guarantee that such a function, even if creatable, be an injection or a surjection.
From: albstorz on 22 Oct 2005 11:19 David R Tribble wrote: > Albrecht Storz wrote: > > So, if there is an infinite set there is an infinite number. > > Do you mean that an infinite set (or natural numbers) must contain an > infinite number as a member (which is false)? Or do you mean that > the size of an infinite set is represented by an infinite number > (which is partially true)? Depending on the axiomatic construction and depending on the necessary of truth (since truth means logic consequence) either there are infinite natural numbers or there is no infinite set. Regards AS
From: albstorz on 22 Oct 2005 11:23 Virgil wrote: > In article <1129946611.902596.262670(a)g49g2000cwa.googlegroups.com>, > "William Hughes" <wpihughes(a)hotmail.com> wrote: > > > Dave Rusin wrote: > > > In article <1129901285.084347.34810(a)g49g2000cwa.googlegroups.com>, > > > William Hughes <wpihughes(a)hotmail.com> wrote: > > > > > > >Yes all sets have a cardinality. > > > > > > That's your Choice. > > > > Am I assuming AC here? > > > > My understanding was that, while AC was needed to > > put a total ordering on the equivalence classes > > under bijection, one could assert the existence > > of these classes even without AC. Or are the > > cardinalities defined not as all possible > > equivalence classes but as the equivalence classes > > on the ordinals? > > > > -William Hughes > > As I understand it, without AC, one cannot be sure of creating any sort > of function at all from one arbitrary set to another, much less > guarantee that such a function, even if creatable, be an injection or a > surjection. Modern math concept leads to garbage thinking. Regards AS
From: albstorz on 22 Oct 2005 11:34 David R Tribble wrote: > Albrecht Storz wrote: > > You can biject all constructable subsets of P(N) to N, but not the > > unconstructable. That's an argument against the diagonal proof, since, > > if a set will be constructable in form of an antidiagonal of a list, > > it's a constructable set and therefore it is element of the set of the > > sets which are bijectable with N. > > Then that constructable set will be in the original list, and the > diagonal set will be different from it, and it can't be a set > constructed from the diagonal. > > But that just proves that you can't construct a set from the > diagonal of the list of sets while allowing that set to also be > in the list. No such set exists in the list. Therefore, the > constructed set must be a set that does not exist in the mapping, > and therefore the mapping does not map every possible set in > the list. And you are shure the nonconstructable exists? And what meaning did they have to you? You like them. Are they nice. What kind of properties they have (elese than be unconstructable)? And what do you do with them then knowing they are? Regards AS
From: albstorz on 22 Oct 2005 11:45
David R Tribble wrote: > William Hughes wrote: > >> Yes every number is a set. No, not every set is a number. For example > >> {peach, apple, plum, fiddle} is a set but not a number. > > > > Albrecht Storz wrote: > >> Wrong. It's a number. The number is this aspect of the set which it > >> have in common with the set {diddle, daddle, doddle, duddle}. > >> To say, a > >> set is a number or a set has a number is a slight difference which has > >> no effect in this considerations. > > Albrecht, if every number is a set, which set is 3? > > What number is the set {a, {b}, {{c}}, {{{d}}}, ...}? > > If sets are numbers, then adding sets should obey the same rules as > adding numbers, right? What is {1,2,3} + {0,2,4}? Is it 3+3 = 6? 3 is every set with 3 members. Shurely the sets are different. And they have different properties. But in the aspect of size of membership they are exact same. Natural numbers are the essence of the sets concerning their size. David,try to be logic and honest to yourself. It's not a play to win or loose. It's a play to understand or to stay weak in mind. You might belief in a one century old tale or in a more than a thousand years old known thruth. Regards AS |