For an uncountable cardinal $\kappa$, we are interested in the least size of a cofinal subset of the partial order $([\kappa]^\omega, \subseteq)$. It is obvious that this cofinality is at least $\kappa$ and a rather simple induction shows that $cof([\aleph_n]^\omega, \subseteq)=\aleph_n$ for each natural number $n \geq 1$.

What is known in $ZFC$ for bigger cardinals?

Also I am intrigued by the following statement made by Alan Dow in Efimov spaces and the splitting number, Topology Proc. (2005):

It is a "large cardinal" hypothesis to assume that there is a cardinal $\kappa$ with uncountable cofinality such that this cofinality is greater than $\kappa$.

What exactly does he mean by that, and what is a good reference to read about it?

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    $\begingroup$ Pcf theory. ${}$ $\endgroup$ – Andrés E. Caicedo Dec 6 '17 at 23:42
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    $\begingroup$ An excellent reference to get started is Eisworth's chapter in the Handbook. $\endgroup$ – Andrés E. Caicedo Dec 7 '17 at 1:11
  • $\begingroup$ Do you understand what are large cardinals and what is a large cardinal hypothesis? $\endgroup$ – Asaf Karagila Dec 7 '17 at 8:23
  • $\begingroup$ Subsection 3.1 of doi.org/10.1016/j.apal.2005.09.012 may be illuminating. $\endgroup$ – saf Dec 7 '17 at 15:03
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    $\begingroup$ @AsafKaragila: I have seen large cardinals and I have seen large cardinal hypothesis, but I don't know a precise definition of either of those concepts. $\endgroup$ – Ramiro de la Vega Dec 8 '17 at 13:31

I think he means the consistency of "there exists a cardinal $\kappa$ with uncountable cofinality so that $cf([\kappa]^\omega, \subseteq) > \kappa$" requires large cardinal axioms.

To motivate this, note that for example if $0^\sharp$ does not exist, then for any cardinal $\kappa > 2^{\aleph_0}$ of uncountable cofinality we have $cf([\kappa]^\omega, \subseteq) = \kappa$.

In fact one can say more. A related argument is given in Theorem 3.1 of Singular cofinality conjecture and a question of Gorelic.

  • $\begingroup$ How about cardinals below the continuum? $\endgroup$ – Ramiro de la Vega Dec 8 '17 at 13:42
  • $\begingroup$ @Ramiro same same. See the proof of Theorem 3.13(d) of doi.org/10.1016/j.apal.2005.09.012 $\endgroup$ – saf Dec 8 '17 at 14:02
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    $\begingroup$ @saf , so it is correct to say that If there is a $\kappa$ of uncountable cofinality for which $cof([\kappa]^\omega,\subseteq)>\kappa$ then there is an inner model with a measurable cardinal? $\endgroup$ – Ramiro de la Vega Dec 8 '17 at 14:20
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    $\begingroup$ @Ramiro correct. $\endgroup$ – saf Dec 9 '17 at 6:57

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