A new definition of Cardinality.
in [Logic]
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From: zuhair on 27 Nov 2009 21:43 On Nov 27, 8:13 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > 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. I was speaking of the opposite. If there is a surjection from the reals to an ordinal, then the reals are well orderable? Zuhair
From: Rupert on 28 Nov 2009 05:07 On Nov 28, 1:43 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Nov 27, 8:13 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > > > 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. > > I was speaking of the opposite. If there is a surjection from the > reals to an ordinal, then the reals are well orderable? > > Zuhair- Hide quoted text - > > - Show quoted text - No that is not true, either. It can be proved in ZF that there is a surjection from the reals to omega. But it cannot be proved in ZF that the reals can be well-ordered unless ZF is inconsistent.
From: zuhair on 28 Nov 2009 06:50 On Nov 27, 8:13 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > 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. Oh yes. But in this way can't we prove in ZF that there would be an injection from the reals to aleph-one. Zuhair
From: zuhair on 28 Nov 2009 06:51 On Nov 28, 5:07 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Nov 28, 1:43 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > > > On Nov 27, 8:13 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > 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. > > > I was speaking of the opposite. If there is a surjection from the > > reals to an ordinal, then the reals are well orderable? > > > Zuhair- Hide quoted text - > > > - Show quoted text - > > No that is not true, either. It can be proved in ZF that there is a > surjection from the reals to omega. But it cannot be proved in ZF that > the reals can be well-ordered unless ZF is inconsistent. Yea, of course, I was confused. You are right. Zuhair
From: Rupert on 28 Nov 2009 13:50
On Nov 28, 10:50 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Nov 27, 8:13 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > > > 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. > > Oh yes. But in this way can't we prove in ZF that there would be an > injection from the reals to aleph-one. > > Zuhair- Hide quoted text - > > - Show quoted text - Yes, correct. But we can't prove that in ZFC either actually, because the continuum may not be that small. But given any set we can take the first ordinal greater than all the ordinals injectable into it. In ZFC that will be the successor of the cardinality of the set and the set will be injectable into it, but in ZF the set is not necessarily injectable into it, as you observe. |