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Is it known if the following question has any answer (possibly under some certain set of axioms) :

How many free ultrafilters there are on $\mathbb{N}$ ?

I am aware that assuming AC, there is at least one.

Thanks. Any reference is highly appreciated :)

valls
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    There are $2^{(2^{|X|})}$ ultrafilters on a set $X$. And there are $|X|$ principal ultrafilters so there are $2^{(2^{|X|})} = P(\mathbb R)$ principal ultrafilters on $\mathbb N$. –  Mar 20 '17 at 10:00
  • @N.H.: If $X$ is finite, there are only $|X|$ ultrafilters. – hmakholm left over Monica Mar 20 '17 at 10:32
  • Sure, I should have precised that $X$ was infinite. –  Mar 20 '17 at 10:36
  • @N.H.: Even when $X$ is infinite, how do you know there are $2^{2^{|X|}}$ different ultrafilters? This doesn't seem obvious to me, even assuming AC. – hmakholm left over Monica Mar 20 '17 at 10:43
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    This is proven page 133 in Counterexample in Topology. The proof was not super obvious if I remember well (I read it long time ago but remembered the result). –  Mar 20 '17 at 10:48
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    http://math.stackexchange.com/questions/83526/the-set-of-ultrafilters-on-an-infinite-set – Asaf Karagila Mar 20 '17 at 11:58
  • Also, without AC, http://math.stackexchange.com/questions/779755/unique-ultrafilter-on-omega – Asaf Karagila Mar 20 '17 at 12:43

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