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Easton's theorem can give a very weak nontrivial constraint on continuum function, but it does not hold for singular cardinals. So:

  1. What are the non-trivial constraints on continuum function in singular cardinals?
  2. Is it possible to well-define a concept of "Continuum function maximum of singular cardinals" by these constraints?
  3. How strong is the consistency strength of this definition?
YCor
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Ember Edison
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1 Answers1

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The intricacies of arithmetic at singular cardinals notwithstanding, I think you're looking for something which doesn't exist.

Given any cardinals $\kappa,\lambda$, singular or regular, there is a (set) forcing extension preserving cardinals and cofinalities in which $2^\kappa>\lambda$. Upper bounds only appear in a meaningful sense when we try to control the value of the continuum function on many inputs simultaneously. For example, famously Shelah proved that $2^{\aleph_\omega}<\aleph_{\omega_4}$ provided that $2^{\aleph_n}<\aleph_\omega$ for all $n<\omega$, but it's perfectly consistent that $2^{\aleph_\omega}\ge\aleph_{\omega_4}$ since for example we could have $2^{\aleph_0}=\aleph_{\omega_{17}+42}$ already.

Noah Schweber
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  • So Can we well-define "the maximization of the growth rate of the continuum function"? – Ember Edison Dec 26 '21 at 06:26
  • @EmberEdison I think once we start asking global questions we'll quickly wind up with open problems. For example, even "$2^\kappa>\kappa^+$ for every infinite cardinal $\kappa$" involves large cardinals. I think some care will need to go into formulating such a question in such a way that it is neither trivial nor ultra-wide-open. – Noah Schweber Dec 26 '21 at 06:44
  • That said, here's one interesting result: it is consistent modulo large cardinals that $2^\kappa$ is weakly inaccessible for every infinite cardinal $\kappa$. See here. If you'd like, I can incorporate these comments into my answer proper. – Noah Schweber Dec 26 '21 at 06:46
  • This doesn't sound like a restriction on continuum function other than the "strongly inaccessible" prohibition. I was also recently discussing with my friend whether "Is there a model about $2^{\kappa}$ is $\kappa$-weakly-inaccessible for every infinite cardinal $\kappa$". – Ember Edison Dec 26 '21 at 10:49
  • So, Do you think we are missing the "puzzle pieces" that make the concept of "continuum function maximum" to absolute, or is there no such concept at all? I feel like I've been reading papers on PCF theory for the past few days and it doesn't really help with this goal... – Ember Edison Dec 26 '21 at 10:53