*EDIT: in retrospect this question should have been split up; I've accepted Joel's answer to the first part below, and asked the second part here.*

*This question is crossposted at MSE; however, it has not gotten any responses there despite a bounty, so I'm asking it here.*

Say that a theory $T$ in the language of ordered fields + constants is *$\mathbb{R}$-satisfiable* if it has a model whose ordered field part is $\mathbb{R}$, with the usual ordered field structure. I'm interested in what compactness-like properties $\mathbb{R}$-satisfiability has.

To begin with, we can easily knock off compactness itself: since $\mathbb{R}$ is Archimedean, $\mathbb{R}$-satisfiability is not compact. But things get murkier when we consider “compactness at higher cardinalities.”

For cardinals $\kappa<\lambda$, say that *$\mathbb{R}$-satisfiability is $(\kappa, \lambda)$-compact* if whenever $\Gamma$ is a set of sentences of cardinality $<\lambda$, and every subset of cardinality $<\kappa$ is $\mathbb{R}$-satisfiable, then $\Gamma$ is $\mathbb{R}$-satisfiable. (So usual compactness is $(\omega, \infty)$-compactness, and countable compactness is $(\omega,\omega_1$)-compactness.)

It’s easy to show that $\mathbb{R}$-satisfiability is not $(\omega_1,\omega_2)$-compact: any countable linear order embeds into $\mathbb{R}$, but $\omega_1$ does not. And it follows from Easton’s theorem that for every cardinal $\kappa$ with $cf(\kappa)>\omega$, *it is consistent with ZFC* that $\mathbb{R}$-satisfiability is not $(\kappa^+, \kappa^{++})$-compact (look at a theory asserting the existence of $\kappa^+$-many distinct reals assuming $\mathfrak{c}=\kappa$).

This suggests two natural questions:

Can ZFC prove that $\mathbb{R}$-satisfiability

*is*$(\omega_{\omega+1}, \omega_{\omega+2})$-compact?Is it

*consistent*with ZFC that $\mathbb{R}$-satisfiability is $(\omega_2, \omega_3)$-compact?

I believe the answer to the second question should be a relatively easy "yes", while the first question should be "no" but might require some work. However, I don't immediately see how to resolve either piece.

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