A new definition of Cardinality.
in [Logic]
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From: zuhair on 24 Nov 2009 20:12 On Nov 24, 7:11 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2009-11-24, George Greene <gree...(a)email.unc.edu> wrote: > > > There are MANY DIFFERENT well-orderings of p(w) under ZFC and they > > correspond TO DIFFERENT ordinals! > > But not to different cardinals. > > - Tim Yes to different *cardinals*, since Von Neumanns' cardinals are special type of ordinals, in ZFC there many different well ordering of p(w) and they correspond to different VON NEUMANN CARDINALS, it can be aleph_1, aleph_2,..., aleph_n were n is is a natural number, all these are CARDINALS, not just ordinals! Zuhair
From: Tim Little on 24 Nov 2009 21:18 On 2009-11-25, zuhair <zaljohar(a)gmail.com> wrote: > Yes to different *cardinals*, since Von Neumanns' cardinals are > special type of ordinals, in ZFC there many different well ordering > of p(w) and they correspond to different VON NEUMANN CARDINALS ZFC proves that only one well-ordering of p(w) has the order-type of a von Neumann cardinal. > it can be aleph_1, aleph_2,..., aleph_n were n is is a natural > number, all these are CARDINALS, not just ordinals! Those are only labels for different properties for card(p(w)) in different models. In any given model, only one of the labels describes card(p(w)), and the rest do not. This does not mean that cardinality is undefined, any more than ring theory leaves the concept of "ideals" undefined because (1+1) = R in some models while (1+1) =/= R in others. - Tim
From: zuhair on 25 Nov 2009 07:33 On Nov 24, 5:53 pm, zuhair <zaljo...(a)gmail.com> wrote: > I got the following reply from an authority on set theory. > > ... Perhaps I misunderstand your definition, but with > Foundation one can prove that every non-empty set contains > 0 (= empty set} in its transitive closure. If I am correct, > then your definition does not work.... > > Can anybody explain that? > > Zuhair The authority himself explained that. There was a misreading of the original definition. Zuhair
From: Marc Alcobé García on 25 Nov 2009 08:44 > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) wrote: > 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. Sorry, may I ask which of Jech's papers is here being referred to? Is it available from http://www.math.psu.edu/jech/preprints/papers.html? Thank you in advance.
From: Aatu Koskensilta on 25 Nov 2009 08:53
Marc Alcob� Garc�a <malcobe(a)gmail.com> writes: > Sorry, may I ask which of Jech's papers is here being referred to? "On hereditarily countable sets". > Is it available from http://www.math.psu.edu/jech/preprints/papers.html? Apparently not. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |