What kind of compactness does "expanding $\mathbb{R}$ by constants" have?

EDIT: Now crossposted at mathoverflow.

This arose from my answer to another question. 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.

Solution 1:

Both questions have negative answers. This was show at mathoverflow by Joel David Hamkins and Farmer S; I've summarized their answers below to move this off the unanswered queue, and I've made this answer CW to avoid reputation gain. Below I use the somewhat more flexible notation introduced in those posts rather than the notation in the OP.

Joel observed that $\mathsf{ZFC}$ proves that $\mathcal{R}$-satisfiability is not $(2^\omega,(2^\omega)^+)$-compact. Clearly for any structure $\mathcal{A}$ we have that $\mathcal{A}$-satisfiability is not $(\vert\mathcal{A}\vert^+,\vert\mathcal{A}\vert^{++})$-compact; in $\mathcal{R}$ we can "remove a $+$" since every real is type-definable in $\mathcal{R}$. More generally, if $\mathcal{A}$ is "pointwise-type-definable" then $\mathcal{A}$-satisfiability is not $(\vert\mathcal{A}\vert,\vert\mathcal{A}\vert^+)$-compact. Since it is consistent with $\mathsf{ZFC}$ that $2^\omega=\omega_{\omega+1}$, it is also consistent with $\mathsf{ZFC}$ that $\mathcal{R}$-satisfiability is not $(\omega_{\omega+1},\omega_{\omega+2})$-compact.

Farmer's proof that $\mathcal{R}$-satisfiability is $\mathsf{ZFC}$-provably not $(\omega_2,\omega_3)$-compact is a bit more involved. The starting point is the observation that if we add a predicate naming the integers, the resulting structure $\mathcal{R}_\mathbb{Z}$ can talk about reals coding countable well-orderings. We can therefore express that the $\omega_2$-many new constant symbols code non-isomorphic countable well-orderings, which is clearly not $\mathcal{R}_\mathbb{Z}$-satisfiable - but any subtheory of size $\omega_1$ is $\mathcal{R}_\mathbb{Z}$-satisfiable. Farmer shows how we can, with a bit more care, avoid using $\mathbb{Z}$ at all.