Exercises 6, 7, and 8 in section 10.4 of Hodges' big model theory textbook contain an outline of a proof of the consistency of the following statement



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> There exists a countable first-order theory $T$ such that $T$ has no saturated models.
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Call this statement $\mathrm{NSat}\_\omega$ and call the analogous statement with no cardinality restriction $\mathrm{NSat}$.


The proof outlined by Hodges uses a result of Woodin's that it is consistent with $\mathrm{ZFC}$ that for every infinite cardinal $\kappa$, $2^\kappa=\kappa^{++}$, but he mentions that this result assumes some large cardinals and in particular 'a supercompact is more than enough'.


Are large cardinals necessary for this result? There are different meanings of a question like this but a typical formulation would be something like:



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> Does $\mathrm{ZFC}$ + 'there exists a transitive model of $\mathrm{ZFC}+\mathrm{NSat}\_\omega$' prove the existence of transitive models of any typical large cardinal axiom? What about $\mathrm{ZFC}+\mathrm{NSat}$?
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Does the statement 'there exists a first-order theory $T$ with no saturated models' have any set theoretic strength?
Unless I'm missing something, if $\vert T\vert+\aleph\_0<\kappa$ and $\kappa^+=2^\kappa$, then we can build a saturated model of $T$ of cardinality $2^\kappa$. So:


* If we want a countable theory with no saturated model, we need GCH to fail everywhere.
* If we want some theory with no saturated model, we need GCH to hold at most boundedly often.


And these principles have rather high consistency strength. A failure of GCH at a singular limit cardinal is in particular a failure of the [**Singular Cardinal Hypothesis**](https://en.wikipedia.org/wiki/Singular_cardinals_hypothesis), the failure of which is known to be equiconsistent with the existence of a $\kappa$ such that $o(\kappa)=\kappa^{++}$. So this gives a lower bound to both principles.


A better analysis of the strength involved, at least for NSat (although I suspect they'll be identical), is [this wonderful MSE answer](https://math.stackexchange.com/a/3186971/28111) by Andres Caicedo (in particular, my previous paragraph is just a rephrasing of his second-to-last paragraph). However, that's from a while ago, so more might be known now.


