Can we raise aleph cardinals to other aleph cardinals?
in [Math]
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From: lite.on.beta on 30 Sep 2008 01:09 What would aleph0 ^ aleph0 be? What would aleph1 ^ aleph1 be? What would cardinality of P^k(N) be? [That is taking power sets k times on naturals. If k=1, then it is P(N) = 2^aleph0 c^alpeh_i is easier since c = 2^aleph_0, therefore c^alpeh_i = 2^aleph_i*aleph0 = 2^aleph_i Can anyone give me a set with cardinality aleph_1? No derivations required.
From: William Elliot on 30 Sep 2008 05:29 On Mon, 29 Sep 2008 lite.on.beta(a)gmail.com wrote: > What would aleph0 ^ aleph0 be? > What would aleph1 ^ aleph1 be? > aleph_1 assuming CH aleph_2 assuming GCH > What would cardinality of P^k(N) be? > > [That is taking power sets k times on naturals. If k=1, then it is > P(N) = 2^aleph0 > aleph_k assuming GCH. Without GCH, it's called beth_k. > c^alpeh_i is easier since c = 2^aleph_0, therefore > c^alpeh_i = 2^aleph_i*aleph0 = 2^aleph_i > > Can anyone give me a set with cardinality aleph_1? > The reals, assuming CH of course. > No derivations required. > What? You don't like math?
From: Aatu Koskensilta on 30 Sep 2008 05:29 William Elliot <marsh(a)hevanet.remove.com> writes: > On Mon, 29 Sep 2008 lite.on.beta(a)gmail.com wrote: > > > Can anyone give me a set with cardinality aleph_1? > > The reals, assuming CH of course. The set of countable ordinals is the obvious example. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, darüber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: amy666 on 30 Sep 2008 02:57 lite wrote : > > What would aleph0 ^ aleph0 be? > What would aleph1 ^ aleph1 be? both are aleph_aleph_0. > > What would cardinality of P^k(N) be? > > [That is taking power sets k times on naturals. If > k=1, then it is > P(N) = 2^aleph0 > > c^alpeh_i is easier since c = 2^aleph_0, therefore > c^alpeh_i = 2^aleph_i*aleph0 = 2^aleph_i > > Can anyone give me a set with cardinality aleph_1? aleph_aleph_0 is the largest. you asked for a set with card aleph_1 ; thats just any uncountable set e.g. cantors set. > > No derivations required. regards tommy1729
From: amy666 on 30 Sep 2008 03:00
> lite wrote : > > > > > What would aleph0 ^ aleph0 be? > > What would aleph1 ^ aleph1 be? > > both are aleph_aleph_0. sorry i read ^^ instead of ^ first is aleph_1 and second is aleph_2. > > > > > > What would cardinality of P^k(N) be? > > > > [That is taking power sets k times on naturals. > If > > k=1, then it is > > P(N) = 2^aleph0 > > > > c^alpeh_i is easier since c = 2^aleph_0, > therefore > > c^alpeh_i = 2^aleph_i*aleph0 = 2^aleph_i > > > > Can anyone give me a set with cardinality aleph_1? > > aleph_aleph_0 is the largest. > > you asked for a set with card aleph_1 ; thats just > any uncountable set e.g. cantors set. > > > > > No derivations required. > > regards > > tommy1729 regards tommy1729 |