A new definition of Cardinality.
in [Logic]
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From: zuhair on 26 Nov 2009 05:39 On Nov 26, 3:22 am, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) writes: > > [Deletion] > > > PS. I have e-mailed you a copy of the Jech paper. I haven't myself > > studied the argument for the existence of the set of hereditarily > > countable sets in absence of choice, by establishing that the rank of a > > hereditarily countable set <= omega_2, and so can say nothing about > > whether it may be generalised beyond the countable case. > > > -- > > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > > "Wovon man nicht sprechen kann, darüber muss man schweigen" > > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > > Thanks. Jech's proof in ZF that HC is a set (not a proper class) > is in the first 2 pages of the paper. > > I think Jech's proof straightdforwardly generalizes to H_kappa > for von Neumann cardinals kappa, showing from ZF that these are sets, > namely replacing omega in Jech's proof by kappa. (Well with > adjustments: HC is heritarily <= omaga and H_kappa is > heritarily < kappa : <= versus < ). > > I also think Jech's proof generalizes to H_(< #x) for > non-well-orderable x, by an adjusted as above replacement > of omega by the surjective Hartog ordinal of x. > > If this is correct, it would show Zuhair's definition of cardinal > is a set for arbtrary x as the theorem of ZF, depending on > regularity and replacement but no AC needed. Yeah, I need to look into that carefully. It appears to agree with my guess that these cardinals I defined are not choice dependent. Thanks David. Zuhair > > -- > David Libert ah...(a)FreeNet.Carleton.CA
From: Jesse F. Hughes on 26 Nov 2009 09:23 George Greene <greeneg(a)email.unc.edu> writes: > NO THERE *DOESN'*, DUMBASS! (DAMN the STUPID is getting > THICK in here)! If I were sitting where you are, I'm sure I'd think the stupid was pretty thick too. Rupert was correct. You act as if he wasn't, but he was, and this is one of those weird situations not solved by the caps lock key. -- Jesse F. Hughes "Really, I'm not out to destroy Microsoft. That will just be a completely unintentional side effect." -- Linus Torvalds
From: zuhair on 27 Nov 2009 07:03 On Nov 26, 3:22 am, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) writes: > > [Deletion] > > > PS. I have e-mailed you a copy of the Jech paper. I haven't myself > > studied the argument for the existence of the set of hereditarily > > countable sets in absence of choice, by establishing that the rank of a > > hereditarily countable set <= omega_2, and so can say nothing about > > whether it may be generalised beyond the countable case. > > > -- > > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > > "Wovon man nicht sprechen kann, darüber muss man schweigen" > > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > > Thanks. Jech's proof in ZF that HC is a set (not a proper class) > is in the first 2 pages of the paper. > > I think Jech's proof straightdforwardly generalizes to H_kappa > for von Neumann cardinals kappa, showing from ZF that these are sets, > namely replacing omega in Jech's proof by kappa. (Well with > adjustments: HC is heritarily <= omaga and H_kappa is > heritarily < kappa : <= versus < ). > > I also think Jech's proof generalizes to H_(< #x) for > non-well-orderable x, by an adjusted as above replacement > of omega by the surjective Hartog ordinal of x. if there is a surjective ordinal on x, then x must be well orderable. Zuhair > > If this is correct, it would show Zuhair's definition of cardinal > is a set for arbtrary x as the theorem of ZF, depending on > regularity and replacement but no AC needed. > > -- > David Libert ah...(a)FreeNet.Carleton.CA
From: K_h on 27 Nov 2009 17:28 "zuhair" <zaljohar(a)gmail.com> wrote in message news:32349214-92a2-4803-9aae-b12de4e0dac6(a)p33g2000vbn.googlegroups.com... On Nov 24, 12:55 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Nov 23, 10:37 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > > > I would like to suggest the following > > > > > > definition: > > > > > > > 4) The cardinality of any set x is: The class of > > > > > > all sets > > > > > > that are equinumerous to x were every member of > > > > > > their transitive > > > > > > closure is strictly subnumerous to x. Here is a fast and easy way to define cardinals. Define the surreal numbers and their order using Conway's simple construction convention and definition (these only require a few lines). Call surreals of the form {x|} "left surreals". Then define the cardinality of a set to be the least left surreal equinumerous to it. You're done! One reason this definition is so effective is because infinite left surreals are equinumerous to all infinite left surreals less than themselves. Additionally, one can distinguish between finite and infinite left surreals by whether or not they can be equinumerous to proper subclasses of themselves. http://planetmath.org/encyclopedia/OmnificIntegers.html
From: Rupert on 27 Nov 2009 20:13
On Nov 27, 11:03 pm, zuhair <zaljo...(a)yahoo.com> wrote: > On Nov 26, 3:22 am, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > > > > > > > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) writes: > > > [Deletion] > > > > PS. I have e-mailed you a copy of the Jech paper. I haven't myself > > > studied the argument for the existence of the set of hereditarily > > > countable sets in absence of choice, by establishing that the rank of a > > > hereditarily countable set <= omega_2, and so can say nothing about > > > whether it may be generalised beyond the countable case. > > > > -- > > > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > > > "Wovon man nicht sprechen kann, darüber muss man schweigen" > > > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > > > Thanks. Jech's proof in ZF that HC is a set (not a proper class) > > is in the first 2 pages of the paper. > > > I think Jech's proof straightdforwardly generalizes to H_kappa > > for von Neumann cardinals kappa, showing from ZF that these are sets, > > namely replacing omega in Jech's proof by kappa. (Well with > > adjustments: HC is heritarily <= omaga and H_kappa is > > heritarily < kappa : <= versus < ). > > > I also think Jech's proof generalizes to H_(< #x) for > > non-well-orderable x, by an adjusted as above replacement > > of omega by the surjective Hartog ordinal of x. > > if there is a surjective ordinal on x, then x must be well orderable. > > Zuhair > > > > > > > If this is correct, it would show Zuhair's definition of cardinal > > is a set for arbtrary x as the theorem of ZF, depending on > > regularity and replacement but no AC needed. > > > -- > > David Libert ah...(a)FreeNet.Carleton.CA- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - This is not true. Let us assume that ZF is consistent. Then ZF can prove that there is a surjection from aleph-one to the reals, but ZF cannot prove that the reals can be well-ordered. |