# Is this compactness property for "satisfiability on $\mathbb{R}$" consistent?

This was originally part of this older question of mine, but in retrospect that question should have been broken into two parts - this is the still-unanswered part.

Let $$\Sigma$$ be the language of ordered fields and let $$\mathcal{R}$$ be the field of real numbers. Say that a theory $$T$$ in a language $$\Sigma'$$ gotten by adding constant symbols to $$\Sigma$$ is $$\mathcal{R}$$-satisfiable iff $$T$$ has a model whose $$\Sigma$$-reduct is $$\mathcal{R}$$ itself. For cardinals $$\kappa<\lambda$$, say that $$\mathcal{R}$$-satisfiability is $$(\kappa,\lambda)$$-compact iff every theory of cardinality $$<\lambda$$ all of whose size-$$<\kappa$$-subsets are $$\mathcal{R}$$-satisfiable is itself $$\mathcal{R}$$-satisfiable.

There are a couple easy observations about the possible extent of compactness for $$\mathcal{R}$$-satisfiability (see the discussion at the above-linked question):

• $$\mathcal{R}$$-satisfiability is provably not $$(\omega_1,\omega_2)$$-compact or $$(2^\omega, (2^\omega)^{+})$$-compact.

• If $$\kappa$$ is measurable then $$\mathcal{R}$$-satisfiability is $$(\kappa,\kappa^+)$$-compact.

However, there is still one natural "low-level" question that remains open:

Is $$\mathcal{R}$$-satisfiability consistently $$(\omega_2,\omega_3)$$-compact?

First, by the above observations $$\mathcal{R}$$-satisfiability can only be $$(\omega_2,\omega_3)$$-compact if $$2^\omega\ge\omega_3$$ (this was pointed out by Joel), so neither $$\mathsf{CH}$$ nor forcing axioms can help us.

A bit more complicatedly, there are subtleties which pose potential obstacles to any "coarse" argument. Specifically, suppose we shift from $$\mathcal{R}$$ to its expansion $$\mathcal{R}_\mathbb{Z}$$ by a predicate naming the integers. In $$\mathcal{R}_\mathbb{Z}$$ we can talk about reals coding countable well-orderings and comparisons between the ordertypes of coded well-orderings. This gives us right off the bat a $$\mathsf{ZFC}$$-provable counterexample to $$(\omega_2,\omega_3)$$-compactness of $$\mathcal{R}_\mathbb{Z}$$-satisfiability: let $$T$$ be the theory using constant symbols $$(c_\eta)_{\eta<\omega_2}$$ saying that the $$c_\eta$$s code well-orderings of distinct ordertypes. This $$T$$ also demonstrates that we cannot obviously focus on complete theories WLOG: while every subtheory of $$T$$ of size $$\omega_1$$ is $$\mathcal{R}_\mathbb{Z}$$-satisfiable, every completion $$S$$ of $$T$$ has some subtheory of size $$\omega_1$$ which is not $$\mathcal{R}_\mathbb{Z}$$-satisfiable since every linear order of size $$\omega_2$$ has a suborder of size $$\omega_1$$ not embedding into $$\omega_1$$. Of course, the above reasoning breaks down for $$\mathcal{R}$$, but this does still suggest that there may be subtleties (and ultimately makes me very skeptical of a positive answer).

It looks to me like under ZFC, $$\mathbb{R}$$-satisfiability is not (consistently) $$(\omega_2,\omega_3)$$-compact. To see this, we'll emulate your argument above for $$\mathbb{R}_{\mathbb{Z}}$$. So basically, we want a theory with constants $$c_\eta$$ for $$\eta<\omega_2$$, which says:

• "$$c_\eta$$ codes a wellorder of $$\omega$$ (in ordertype $$\geq\omega$$)",

• "if $$\eta_0\neq\eta_1$$ then $$c_{\eta_0},c_{\eta_1}$$ have distinct ordertypes".

It suffices to express these things in an appropriate fashion. We will in fact use a bunch more constants to help with this. We start with constants $$\left_{i<\omega}$$, along with the statements:

• "$$n_0=0$$"

• "$$n_1=1$$"

• "$$n_{k+1}=n_k+1$$", for each $$k<\omega$$

So whenever all these formulas are included, $$n_k$$ must be interpreted by $$k$$. Thus, we can talk about the integers by referencing these constants. But we cannot quantify directly over the integers. But we can do this enough for our purposes, indirectly. For each tuple $$\vec{a}=(a_0,\ldots,a_{k-1})$$ of constants that we add, we add further new constants $$t_{\vec{a}}$$ and $$t^{\mathrm{wit}}_{\vec{a}}$$; $$t_{\vec{a}}$$ will code the $$\Sigma_1^{\mathbb{N},\vec{a}}$$ theory (by which I really mean that $$\Sigma_1^{\mathbb{N},A_{\vec{a}}}$$ theory, where $$A_{\vec{a}}=(A_{a_0},\ldots,A_{a_{k-1}})$$ denotes the tuple of sets of integers coded by $$\vec{a}$$, and the theory is the collection of all $$\Sigma_1$$ truths in integer parameters over the structure $$(\mathbb{N},A_{a_0},\ldots,A_{a_{k-1}})$$), and the (interpretation of the) constant $$t^{\mathrm{wit}}_{\vec{a}}$$ will code witnesses to the $$\Sigma_1$$ assertions in this theory. Because our constants are closed under this, we end up with constants which will code the $$\Sigma_k^{\mathbb{N},\vec{a}}$$ theory, for each $$k<\omega$$, via which we can make arithmetical statements, which will be very helpful.

So, for specificity, let's say that the real $$x$$ codes the set of rationals $${, and this then gives us a natural way of coding a set $$A_x$$ of integers with $$x$$. Fix also such a natural way of coding total functions $$f_x:\omega\to\omega$$ with $$x$$. (For example, take $$f_x(0)=$$ the floor of $$x$$, and if $$f_x(0)=i$$, then break the interval $$[i,i+1)$$ into the sub-intervals $$I_0=[i+\frac{1}{2})$$, $$I_1=[i+\frac{1}{2},i+\frac{3}{4})$$, etc, and $$f_x(1)=$$ the $$k$$ such that $$x\in I_k$$, and then break $$I_k$$ up in this manner to define $$f_x(2)$$, etc.)

Note that for each specific tuple $$\vec{n}$$ of integers, we can express $$\Sigma_0^{\mathbb{N},\vec{a}}(\{\vec{n}\})$$ statements with formulas in our theory, by converting them into quantifier-free statements (i.e. convert "$$\exists m" as a disjunction of $$n_0$$-many formulas, etc). Fix an enumeration $$\left<\psi_i\right>_{i<\omega}$$ of all $$\Sigma_0$$ formulas in the language of $$\mathbb{N}$$ plus a predicate (implicit below, interperted as $$A_\vec{a}$$). We include the following statements in our theory:

• "If the formula $$\exists k\psi_i(k,\vec{n})$$' is in $$A_{t_{\vec{a}}}$$, and $$f_{t_{\vec{a}}^{\mathrm{wit}}}(i)=k'$$, as witnessed by calculation $$c$$, then $$\psi_i(k',\vec{n})$$" (here $$i,\vec{n},k',c$$ are arbitrary (tuples of) integers, and we have some fixed coding of "calculations" by integers; note that "$$\psi_i(k',\vec{n})$$", for these particular integers $$k',\vec{n}$$, is expressible as noted above).

• "If the formula $$\exists k\psi_i(k,\vec{n})$$' is not in $$A_{t_{\vec{a}}}$$, then $$\neg\psi_i(k',\vec{n})$$" (here $$i,k',\vec{n}$$ are arbitrary integers).

If we have all these formulas present in the theory, then $$t_{\vec{a}}$$ must be interpreted by (some real coding) the true $$\Sigma_1^{\mathbb{N},\vec{a}}$$-theory. Thus (and as mentioned above) we can make arithmetical statements about the (sets coded by the) constants.

We now want to express "$$c_\eta$$ codes a wellorder of $$\omega$$ (of ordertype $$\geq\omega$$)". We can do this by saying it codes a linear order which is comparable with all countable ordinals. Toward this, fix surjections $$f:\omega\to\alpha$$, for each $$\alpha\in[\omega,\omega_1)$$. We introduce new constants $$d_\alpha$$ for $$\alpha\in[\omega,\omega_1)$$, and, along with $$c_\eta$$, new constants $$\pi_{\eta\alpha}$$ for $$\alpha\in[\omega,\omega_1)$$. We want $$d_\alpha$$ to code a wellorder of ordertype $$\alpha$$, via $$f_\alpha$$, and $$\pi_{\eta\alpha}$$ to be an isomorphism comparing $$c_\eta$$ with $$d_\alpha$$. So add the following statements which achieve this:

• "$$(m,n)\in A_{d_\alpha}$$" (for integers $$m,n$$ and $$\omega\leq\alpha<\omega_1$$ such that $$f_\alpha(m)),

• "$$(m,n)\notin A_{d_\alpha}$$" (for $$m,n,\alpha$$ as above but with $$f_\alpha(m)\geq f_\alpha(n)$$",

• "$$c_{\eta}$$ codes a linear order $$<_\eta$$ of $$\omega$$" (for all $$\eta$$),

• "$$\pi_{\eta\alpha}$$ codes an isomorphism, either from $$c_\eta$$ onto an initial segment of $$d_\alpha$$, or from an initial segment of $$c_\eta$$ onto $$d_\alpha$$" (all $$\eta,\alpha$$).

Note these are each arithmetic statements about the relevant parameters. Finally, we want to say that the ordertypes of the $$c_\eta$$ are pairwise distinct. This is similar. We use further constants $$\sigma_{\eta\beta}$$ for $$\eta<\beta<\omega_2$$, and say:

• "$$\sigma_{\eta\beta}$$ codes an isomorphism, either from $$<_\eta$$ onto a proper segment of $$<_\beta$$, or from a proper segment of $$<_\eta$$ onto $$<_\beta$$" (for each $$\eta<\beta<\omega_2$$).

Now as in your $$\mathbb{R}_{\mathbb{Z}}$$ example, every sub-theory of size $${<\omega_2}$$ is $$\mathbb{R}$$-satisfiable, but the whole thing is not, because any realization is really "correct", i.e. the constants get the desired kinds of meanings.

Update: Re the further question in the comments on $$(\omega_3,\omega_4)$$: Assume ($$\dagger$$) ZFC + for every real $$x$$, $$x^\#$$ exists, and $$u_2=\omega_2$$ (where $$u_\alpha$$ is the $$\alpha$$th uniform indiscernible). Then $$\mathbb{R}$$-satisfiability is not $$(\omega_3,\omega_4)$$-compact.

Remark: Woodin has shown that ($$\dagger$$) is consistent relative to ZFC + a Woodin cardinal + a measurable above it. Moreover, starting with a model $$V\models(\dagger)$$, we can force over $$V$$ by adding a sequence $$G$$ of Cohen reals, hence arranging $$2^\omega$$ as large as we want, and preserve ($$\dagger$$). (For $$u_2$$ can only increase by forcing in general, and $$u_2\leq\omega_2$$ in general. But $$\omega_n^{V[G]}=\omega_n$$, and it follows that $$V[G]\models u_2=\omega_2$$. So it suffices to see that all reals still have sharps. But every real in $$V[G]$$ is added by a single Cohen forcing, and Cohen forcing preserves the existence of sharps for all reals.) So it's not that $$(\omega_3,\omega_4)$$-compactness is necessarily failing due to small continuum.

So assume ($$\dagger$$). The failure of $$(\omega_3,\omega_4)$$-compactness is arranged along the lines of the preceding argument. Define the equivalence relation $$=^*$$ on the reals by $$x=^*y$$ iff $$\kappa_x=\kappa_y$$, where $$\kappa_x$$ is the least $$x$$-indiscernible $$\kappa$$ such that $$\kappa>\omega_1$$. By ($$\dagger$$), $$\{\kappa_x\bigm|x\in\mathbb{R}\}$$ is cofinal in $$\omega_2$$, so there are exactly $$\omega_2$$-many equivalence classes. So let $$\left_{\eta<\omega_3}$$ be some constants. It suffices to see we can express the statements

• "$$c_\eta\neq^* c_\beta$$",

for each $$\eta<\beta<\omega_3$$ (with the help of further constants). Given any constant $$x$$ we introduce, we will also introduce $$s(x)$$, which will be interpreted as $$x^\#$$. For all constants $$s,t$$, we will also introduce a constant $$p(s,t)$$ for $$s\oplus t$$. As before, we also introduce the $$n_k$$'s and close under the $$\Sigma_1$$-theories, so that we can make arithmetic statements. Now the main issue is to ensure that the interpretation of $$s(x)$$ is really $$x^\#$$: given this, the assertion "$$c_\eta\neq^* c_\beta$$" is equivalent to "$$L[s(p(c_\eta,c_\beta))]\models c_\eta\neq^* c_\beta$$" (where we define $$=^*$$ in this "model" $$M$$ as in $$V$$, except that we use $$\omega_1^M$$ instead of $$\omega_1^V$$), which will say that $$L[(c_\eta\oplus c_\beta)^\#]\models$$"$$c_\eta\neq^* c_\beta$$"; note that we can check this by just checking that $$s(s(p(c_\eta\oplus c_\beta)))$$ contains the right statement.

So we want to ensure that $$s_x$$ really gets interpreted as $$x^\#$$. We add further constants: Add $$d_\alpha$$ for $$\omega\leq\alpha<\omega_1$$ as before (coding a wellorder $$<_\alpha$$ of $$\omega$$ of ordertype $$\alpha$$), and $$s_{x,\alpha}$$ (to code the model of the form "$$L_\gamma[x]$$" (as of yet possibly illfounded) one gets generated from an $$\alpha$$-sequence of "$$x$$-indiscernibles" as determined by $$s_x$$), for $$\alpha<\omega_1$$, and $$\pi_{x,\alpha,\beta}$$, to code an isomorphism which compares the ordinals of $$s_{x,\alpha}$$ with $$\beta$$, for each $$\alpha,\beta<\omega_1$$. We add formulas asserting:

• "$$s_{x}$$ is a pre-$$x$$-sharp" (meaning that $$s_x$$ has the right syntactic properties for a sharp, but ignoring iterability / wellfoundedness of the models generated as above),

• "$$(m,n)\in A_{d_\alpha}$$" or "$$(m,n)\notin A_{d_\alpha}$$" (as before),

• "$$s_{x,\alpha}$$ codes the term model generated by an $$\alpha$$-sequence of indiscernibles modelling the theory given by $$s_x$$ (and using $$<_\alpha$$ to determine the indiscernible sequence)",

• "$$\pi_{x,\alpha,\beta}$$ codes either (i) an isomorphism from the ordinals of $$s_{x,\alpha}$$ onto an initial segment of $$<_\beta$$, or (ii) an isomorphism from an initial segment of the ordinals of $$s_{x,\alpha}$$ onto $$<_\beta$$".

These are all arithmetic statements about these constants, so we can make them. And they together ensure the correctness of $$s_x$$, because it ensures that the model $$L_\gamma[x]$$ mentioned above (which is countable) is wellfounded.

As mentioned above, since there are $$\omega_2$$ distinct classes with respect to $$=^*$$, but not $$\omega_3$$, this gives a failure of $$(\omega_2,\omega_3)$$-compactness for $$\mathbb{R}$$-satisfiability.

Thus, what if we work in $$V=L[G]$$ where $$G$$ adds $$\geq\omega_4^L$$ Cohen reals to $$L$$? Does $$L[G]\models$$"$$\mathbb{R}$$-satisfiability is $$(\omega_3,\omega_4)$$-compact"?

And however, I think $$(\omega_4,\omega_5)$$ will be more subtle.

Re $$(\kappa,\kappa^{++})$$-compactness for $$\mathbb{R}$$-satisfiability, don't we get that from $$\kappa^+$$-supercompactness of $$\kappa$$, like with the measurability giving $$(\kappa,\kappa^+)$$? (And likewise for $$(\kappa,\lambda)$$-compactness, if $$\kappa$$ is $$<\lambda$$-supercompact.) I.e. if $$T$$ is a theory of size $$\gamma\in[\omega,\lambda)$$, and all subtheories of size $$<\kappa$$ are $$\mathbb{R}$$-satisfiable, and $$j:V\to M$$ is elementary with $$\gamma and $${^\gamma}M\subseteq M$$, then $$M$$ thinks "all subsets of $$j(T)$$ of size $${ are $$\mathbb{R}$$-satisfiable". But $$jT\in M$$ and $$jT$$ has size $$\gamma in $$M$$, so $$jT$$ has an $$\mathbb{R}$$-model $$\mathbb{R}^+\in M$$, but note $$\mathbb{R}^+\models T$$. (Of course this actually has nothing to do with $$\mathbb{R}$$; it could have been any structure of size $${<\kappa}$$.)

Update 2: Although it cannot be $$(\omega_2,\omega_3)$$-compact, it turns out that, relative to ZFC + a weakly compact, $$\mathcal{R}$$-satisfiability can be $$(\kappa,\kappa^+)$$-compact with uncountable $$\kappa$$, even with large continuum, i.e. $$2^{\aleph_0}>\kappa$$. (On the other hand, in connection with the foregoing argument, and this question https://math.stackexchange.com/questions/4166734/does-mathsfzfc-prove-that-the-field-of-real-numbers-has-one-of-these-compac/4172264#4172264 and its answers, a reasonable question was whether every uncountable cardinal $$\kappa$$ such that $$\mathcal{R}$$-satisfiability is $$(\kappa,\kappa^+)$$-compact, has to be weakly compact.) It turns out that the most obvious candidate for a (counter)example works: Suppose $$\kappa$$ is weakly compact in $$L$$. Let $$G$$ be $$L$$-generic for adding a $$\kappa^+$$-sequence of Cohen reals with finite support. Then in $$L[G]$$, $$\mathcal{R}$$-satisfiability is $$(\kappa,\kappa^+)$$-compact, but $$2^{\aleph_0}=\kappa^+$$, so $$\kappa$$ is not weakly compact.

Proof: Work in $$V=L[G]$$. Let $$T$$ be a relevant theory of size $$\kappa$$, all of whose size $${<\kappa}$$ sub-theories are $$\mathcal{R}$$-satisfiable. We want to see $$T$$ is also. Note that there is $$\alpha<\kappa^+$$ such that $$T\in L_{\alpha}[G\upharpoonright\alpha]$$, and so by rearranging $$G$$ a little, we may assume that $$T\in L_\alpha[G\upharpoonright\kappa]$$ for some $$\alpha<\kappa^+$$. Let $$X\preccurlyeq L_{\kappa^{++}}$$ with $$\beta=X\cap\kappa^+$$ transitive and $$\alpha<\beta<\kappa^+$$ and $$|X|=\kappa$$. Let $$L_\gamma$$ be the transitive collapse of $$X$$. Using the weak compactness of $$\kappa$$ in $$L$$, let $$\xi<\kappa^+$$ and $$j:L_{\gamma}\to L_\xi$$ be elementary with $$\mathrm{crit}(j)=\kappa$$. So $$\beta. Note that $$G\upharpoonright j(\kappa)$$ is $$L_\xi$$-generic and extends $$G\upharpoonright\kappa$$, and we can extend $$j$$ elementarily to $$j^+:L_\gamma[G\upharpoonright\kappa]\to L_\xi[G\upharpoonright j(\kappa)]$$. Now by the homogeneity of the forcing, $$L_{\kappa^{++}}[G\upharpoonright\kappa]\models$$"$$\mathbb{P}$$ forces that $$T$$ is $${<\kappa}$$-$$\mathcal{R}$$-satisfiable", where $$\mathbb{P}$$ adds $$\kappa^+$$-many Cohen reals with finite support (and the "$$\mathcal{R}$$" refers to the $$\mathcal{R}$$ of the $$\mathbb{P}$$-extension). So $$L_\gamma[G\upharpoonright\kappa]$$ models the same, but with $$\mathbb{P}$$ replaced by $$\mathbb{P}\upharpoonright\beta$$. So $$L_\xi[G\upharpoonright j(\kappa)]$$ models the same regarding $$j(T)$$ and $$j(\kappa)$$ and $$\mathbb{P}\upharpoonright j(\beta)$$. We have $$j(T)\upharpoonright\kappa=T$$, so $$L_\xi[G\upharpoonright j(\kappa)]\models$$"$$\mathbb{P}\upharpoonright j(\kappa)$$ forces that $$T$$ is $$\mathcal{R}$$-satisfiable". So $$L_\xi[G\upharpoonright j(\beta)]\models$$"$$T$$ is $$\mathcal{R}$$-satisfiable". Let $$\pi:T\to\mathbb{R}^{L_\xi[G\upharpoonright j(\beta)]}$$ be an $$\mathcal{R}$$-realization of $$T$$ in $$L_\xi[G\upharpoonright j(\beta)]$$. So $$\pi\in L[G]$$, and so it suffices to see that $$\pi$$ is also an $$\mathcal{R}$$-realization of $$T$$ in $$L[G]$$. For this, it suffices to see that $$\mathcal{R}^{L_\xi[G\upharpoonright j(\beta)]}\preccurlyeq\mathcal{R}^{L[G]}$$. But the theory of real closed fields has an algorithm for quantifier elimination, which both $$L[G]$$ and $$L_\xi[G\upharpoonright j(\beta)]$$ agree about, and since $$\mathcal{R}^{L_\xi[G\upharpoonright j(\beta)]}$$ is a sub-field (real closed) of $$\mathcal{R}^{L[G]}$$, this fact yields the desired elementarity.

A straightforward variant of this also works in $$L[G]$$ for any sequence $$G$$ of Cohen reals of length $$\geq\kappa^+$$.

• Oh I like this a lot! Two quick questions. First, this does not obviously help with questions like "Is it consistent that $2^{\omega}\ge\omega_4$ and $\mathcal{R}$-satisfiability is $(\omega_3,\omega_4)$-compact?," right? I think this is fairly special to the $(\omega_2,\omega_3)$-case, but I just want to check that I'm not missing something. Second, do you think it's plausible that there is some $\kappa$ such that $\mathcal{R}$-satisfiability is $(\kappa,\kappa^{++})$-compact? Jun 8, 2021 at 2:50
• @NoahSchweber I added some more on these questions. Jun 8, 2021 at 15:59
• Re: your final addition, for the "gap-two" stuff I'm most interested in what can be done in $\mathsf{ZFC}$ alone (or very mild strengthenings thereof), so while supercompactness works it's not what I'm looking for there. Jun 9, 2021 at 3:12
• Separately, could you say a bit about why you think the $(\omega_4,\omega_5)$-case may be particularly subtle? Jun 9, 2021 at 3:12
• @NoachSchweber I added Update 2, relating to your questions on $(\kappa,\kappa^+)$-compactness with large continuum. The position of my statement "I think $(\omega_4,\omega_5)$ will be more subtle" in the text might have been misleading: I meant I thought it would be more subtle to do an argument analogous to those for $(\omega_2,\omega_3)$ and $(\omega_3,\omega_4)$, not the $L[G]$ question. This is because (i) $u_{n+3}$ is singular, so $u_{n+3}\neq\omega_{n+3}$, and (ii) if there was something like it, it should lead into larger cardinals. But this is moot now given the recent answers on MSE. Jun 17, 2021 at 1:29