Aleph_0 ^ Aleph_0
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From: David C. Ullrich on 12 Jan 2010 10:45 On Tue, 12 Jan 2010 04:47:44 -0800, William Elliot <marsh(a)rdrop.remove.com> wrote: >On Tue, 12 Jan 2010, Tonico wrote: >> On Jan 12, 2:32�pm, Leonid Lenov <leonidle...(a)gmail.com> wrote: > >>> How much is Aleph_0 ^ Aleph_0? Is it equal to Aleph_1? > >Yes, assuming the continuum hypothesis. > >> Put A:= Aleph_o, c = the continuum cardinality,then: >> >> c = 2^A <= A^A <= (2^A)^A = 2^(A^2) = 2^A = c >> >> By Bernstein-Schroeder's theorem, A^A = c > >Theorem. GCH. >Proof. Occam's Razor. Uh, right.
From: Butch Malahide on 12 Jan 2010 11:55 On Jan 12, 8:33 am, A N Niel <ann...(a)nym.alias.net.invalid> wrote: > In article > <6f00e4d9-ae47-4a4a-97d3-1d245d84e...(a)c3g2000yqd.googlegroups.com>, > > Leonid Lenov <leonidle...(a)gmail.com> wrote: > > If we do not assume the continuum hypothesis is Tonico's argument > > still valid to show that Aleph_0 ^ Aleph_0 <= Aleph_1? > > No, in ZF it shows aleph_0 ^ aleph_0 = c >= aleph_1 . By "ZF" you must mean ZFC, standard set theory *including* the axiom of choice, right? The abbreviation "ZF" usually refers to choiceless set theory. Of course you need the axiom of choice to prove that the cardinality of the continuum is comparable with aleph_1.
From: A N Niel on 12 Jan 2010 14:17 In article <1eea14fb-2d9f-4ccb-8a7c-ea462ec9db7c(a)k17g2000yqh.googlegroups.com>, Butch Malahide <fred.galvin(a)gmail.com> wrote: > On Jan 12, 8:33�am, A N Niel <ann...(a)nym.alias.net.invalid> wrote: > > In article > > <6f00e4d9-ae47-4a4a-97d3-1d245d84e...(a)c3g2000yqd.googlegroups.com>, > > > > Leonid Lenov <leonidle...(a)gmail.com> wrote: > > > If we do not assume the continuum hypothesis is Tonico's argument > > > still valid to show that Aleph_0 ^ Aleph_0 <= Aleph_1? > > > > No, in ZF it shows aleph_0 ^ aleph_0 = c >= aleph_1 . > > By "ZF" you must mean ZFC, standard set theory *including* the axiom > of choice, right? The abbreviation "ZF" usually refers to choiceless > set theory. Of course you need the axiom of choice to prove that the > cardinality of the continuum is comparable with aleph_1. My mistake. I mean ZFC. But, the point, without CH.
From: Math1723 on 12 Jan 2010 14:28 On Jan 12, 7:32 am, Leonid Lenov <leonidle...(a)gmail.com> wrote: > Hello, > How much is Aleph_0 ^ Aleph_0? Is it equal to Aleph_1? > Thanks in advance. Assuming ^ means cardinal exponentiation, Aleph_0 ^ Aleph_0 = c where c is the cardinality of the continuum. Whether or not c = Aleph_1 is determined by whether you assume the Continuum Hypotheisis or not. Since C.H. is undecidable, this question cannot be answered without more information about C.H.
From: scattered on 12 Jan 2010 15:17
On Jan 12, 2:28 pm, Math1723 <anonym1...(a)aol.com> wrote: > On Jan 12, 7:32 am, Leonid Lenov <leonidle...(a)gmail.com> wrote: > > > Hello, > > How much is Aleph_0 ^ Aleph_0? Is it equal to Aleph_1? > > Thanks in advance. > > Assuming ^ means cardinal exponentiation, Aleph_0 ^ Aleph_0 = c where > c is the cardinality of the continuum. Whether or not c = Aleph_1 is > determined by whether you assume the Continuum Hypotheisis or not. You mean I get to determine how big the continuum is? I didn't realize I had such power. More seriously, I think that it is more accurate to say that the size of the continuum is an open problem but that various possibilities are all consistent with ZFC. c is what c is. If it is in fact Aleph_1 but you assume that it is greater than Aleph_1 then you have made a false assumption, even if ZFC is not powerful enough to prove its falseness. > Since C.H. is undecidable, this question cannot be answered without > more information about C.H. |