2

Is it possible to develop model theory for models of $PA$, inside $PA$ itself (augmented with consistency raising assumptions such as $Con(ZFC)$ if necessary, but still in the language of $PA$)? What would this look like?

When working in ZFC, it is relatively easy to talk about $\mathbb{N}$ as a model of $PA$, because it is an object in the theory, but in $PA$ there is no such object, and the truth predicate for formulas of $PA$ cannot be defined. So it seems like model theory can't really even get off the ground. So what is "the finitist's answer" to model theory as a field of mathematics?

Mario Carneiro
  • 28,221
  • 2
    Note that there's nothing really special about model theory here: PA can't directly treat infinite structures in general, so no mathematics treating infinite entities goes through "directly" in PA. For example, how should PA understand the Heine-Borel theorem? – Noah Schweber Nov 18 '19 at 04:27
  • @NoahSchweber There is a decent literature on how much of analysis can be done "indirectly" in PA. You get a nice language by moving to conservative second order extensions like ACA0, and then you can talk about various kinds of infinite structures as second order objects. For example, https://www.andrew.cmu.edu/user/avigad/Papers/elementary.pdf talks about doing analysis in EA, which is a pretty weak subsystem of PA. – Mario Carneiro Nov 18 '19 at 04:35
  • But that all applies equally well to model theory - in particular, ACA$_0$ proves the completeness theorem and the bivalence of truth. Uncountable structures aren't directly treatable in ACA$_0$ but we can just repeat the same trick and go to yet another conservative extension (cf. "higher-order reverse mathematics"). What new difficulty does model theory pose? – Noah Schweber Nov 18 '19 at 04:35
  • @Noah It's not clear to me how to define the truth predicate of PA, as I mentioned. It's not hard to extend the language to higher order things (I suspect that you eventually end up with some kind of constructive type theory if you take this approach to its conclusion), and provability in these extensions is compensated by additional consistency strengthening in the original base theory $PA$. As far as I am aware this handles the technical restrictions, but it would still be helpful to see what the overall process looks like and what comes out at the end. – Mario Carneiro Nov 18 '19 at 04:43
  • ... Model theory is an example, used because it is easily accessible from a proof theory point of view and has some obviously interesting theorems that go up against Godel's theorems without care. – Mario Carneiro Nov 18 '19 at 04:44
  • 1
    I don't know what you mean by "the truth predicate of PA." A theory doesn't have a truth predicate, a structure does. In the language of second-order arithmetic, "$\models$" has an obvious $\Sigma^1_1$ definition ("There is a family of Skolem functions such that ...") and ACA$_0$ can prove many basic facts about it. For example, ACA$_0$ can directly state and prove "If PA has a model then PA + $\neg$Con(PA) has a model." – Noah Schweber Nov 18 '19 at 04:45
  • @NoahSchweber Aha, I didn't realize that this "skolem function" definition works. If you could expound on that with references or an answer I would like that. (In particular, what PA statement does it correspond to, and what are the basic facts provable in PA/ACA0, and what is needed to prove the facts that we care about, such as soundness and completeness?) – Mario Carneiro Nov 18 '19 at 04:51
  • Sure, will do (although I vaguely recall answering this question before - let me see if I can find it ...). – Noah Schweber Nov 18 '19 at 04:52
  • So I found what I was looking for; it doesn't really answer this question, but I think it's still relevant so I'm going to link it here. – Noah Schweber Nov 18 '19 at 05:05

1 Answers1

2

This old answer of mine is related.

The usual approach - of shifting to an appropriate conservative extension - works fine here. The key observation is that a statement of the form "$M\models\varphi$" is best understood as an existential quantification over objects one type higher than $M$ (namely, "there is a family of Skolem functions for $\varphi$ in $M$").

In particular, this means that $ACA_0$ - the usual go-to expansion of PA - can directly talk about countable model theory: the structures we look at are coded by sets of natural numbers, and "$\models$" is a $\Sigma^1_1$ relation. $ACA_0$ is strong enough to prove basic facts about model theory under this approach; in particular, it proves compactness/completeness (this actually only takes $WKL_0$, which is of strictly weaker consistency strength) and "weak" bivalence: the scheme consisting of, for each sentence $\varphi$, the sentence "For each $M$ we have $M\models\varphi$ or $M\models\neg\varphi$." On the other hand, there are some basic principles $ACA_0$ can't prove:

  • "Strong" bivalence - the single sentence "For every sentence $\varphi$ and every structure $M$ we have $M\models\varphi$ or $M\models\neg\varphi$" - is equivalent over $RCA_0$ to the statement "For every $X,n$, the $n$th jump of $X$ exists," which is strictly stronger than $ACA_0$. (If I recall correctly this theory is denoted "$ACA_0^*$.") A key point here is the computability-theoretic analysis of Skolem functions: we show that we can build Skolem functions uniformly from the appropriate number of Turing jumps.

  • "Every structure has a theory" is even stronger: it's equivalent over $RCA_0$ to $ACA_0^+$, which is $RCA_0$ + "For every $X$, the $\omega$th jump of $X$ exists." (The point is that $Th(\mathbb{N};+,\cdot,X)$ "is" just $X^{(\omega)}$.)

And of course there's the fact that this still doesn't let us handle uncountable structures directly, which are important. But that's also an issue with mathematics in general - we just need to look for higher-order conservative extensions (like higher reverse mathematics' $RCA_0^\omega+\mathcal{E}_1$).

At this point it's worth mentioning some work of Victor Harnik:

  • In $1985$, Harnik studied the reverse math of some theorems of model theory ...

  • but in $1987$ he passed to a richer language to handle uncountable structures.

Barring some unpublished notes of Harvey Friedman from the late $70$s, I believe this was the earliest work in higher-order reverse mathematics; however, it was not followed up on at the time to the best of my knowledge, and the modern approach looks rather different.

Noah Schweber
  • 260,658
  • This brings something I had not realized was important to the fore: ACA${}_0$ is not a definitional extension of PA, it is "merely" a conservative extension, and in particular formulas with existential quantifiers cannot be rendered in PA. Since the formula $M\vDash\varphi$ in question is $\Sigma_1^1$, it can be rendered as $\exists F,\psi(F,M,n)$ where $n$ is a code for $\varphi$ and $M$ and $F$ are second order variables, which in PA is equivalent to the existence of a definable formula $F(a)$ such that $\psiF,M$. But this means PA can only express $M\vDash_F\varphi$ for fixed $M,F$... – Mario Carneiro Nov 18 '19 at 05:51
  • @MarioCarneiro But that's true, for exactly the same reason, of all infinite objects. Again, what singles model theory out here? – Noah Schweber Nov 18 '19 at 05:53
  • I'm fine with the fact that this is schematic over $M$ and $F$, but I'm trying to work my head around the existential part. What would a theorem using the relation look like? You mention weak bivalence: that seems to mean that you are proving $(M\vDash_F\varphi)\lor(M\vDash_G\neg\varphi)$ where $F$ and $G$ are specific expressions in terms of $M$ and $\varphi$? – Mario Carneiro Nov 18 '19 at 05:57
  • 1
    @MarioCarneiro No, the instance of weak bivalence corresponding to a given $\varphi$ is "For all $M$, either $M\models\varphi$ or $M\models\neg\varphi$" - or in your notation, "$\forall M(\exists F(M\models_F\varphi)\vee\exists G(M\models_G\varphi))$." This is a sentence in the language of second-order arithmetic. But again, we have to do this linguistic shift any time we talk about infinite objects; there's nothing special about model theory here. – Noah Schweber Nov 18 '19 at 06:20
  • ACA0 is conservative over PA, but I'm trying to figure out more syntactically what the content of that assertion is in order to get a definition in PA itself. $M\vDash_F\varphi$ says that $F$ is a set of skolem functions validating that $M\vDash\varphi$. I'm trying to figure out what theorem or set of theorems of PA correspond to this assertion in second order logic. I think that it says that given any $M$, you can provide $F$ and $G$ such that $PA\vdash M\vDash_F\varphi\lor M\vDash_G\neg\varphi$, but I could be wrong. I looked at Simpson's book but the conservativity proof is nonconstructive. – Mario Carneiro Nov 18 '19 at 06:23
  • I think looking for a "PA-counterpart" in the first place misses the point of passing to a conservative extension: we're trying to gain expressive power. Why should anything we're talking about in this newer language have a PA-counterpart, whatever that means? (And again this isn't special to model theory - e.g. what's the PA-counterpart of "$[0,1]$ is compact"?) – Noah Schweber Nov 18 '19 at 06:43
  • Also, I don't know what you mean when you say that the conservativity proof is non-constructive. It's completely constructive: if $A$ is a model of PA then $(A,Def(A))$ is a model of ACA$_0$. – Noah Schweber Nov 18 '19 at 06:44
  • What I mean is that it's a model theoretic proof rather than a proof theoretic proof, so it is difficult to see how to turn a proof of $ACA_0\vdash\varphi$ where $\varphi$ is arithmetic into a proof of $PA\vdash\varphi$, and what kind of damage is done to the proof in the process. I found https://mathoverflow.net/questions/127080/what-metatheory-proves-mathsfaca-0-conservative-over-pa indicating that the proof grows superexponentially, but I haven't managed to track down the sources for this yet. – Mario Carneiro Nov 18 '19 at 06:50
  • @MarioCarneiro But that issue is irrelevant here since our $\varphi$s aren't arithmetic (that's sort of the point). – Noah Schweber Nov 18 '19 at 06:50
  • Right, it doesn't totally answer the question, my hope was actually to track down the sources and see what they have to say about the non-arithmetic statements. At the risk of tipping my hand as to my motivation, in my work I am interested in PA extended with schematic variables, which gives it essentially the expressive power of ACA_0 without the existential second order variables. I suspect that model theory such as this is still accessible, but you would have to define some specific terms $F[M]$ and $G[M]$ and prove bivalence with respect to them. – Mario Carneiro Nov 18 '19 at 06:57
  • Regarding "we're trying to gain expressive power", I don't really think of it that way. I just want to circumvent the pedant that says "you can't define the set of prime numbers in PA" when $n\in \mathbb{P}$ is perfectly well definable; I just want a language that puts "sugar over PA" so that it's not a ridiculous question to have PA "talk about infinite sets" that are just formulas with one free variable coding subsets of $\mathbb{N}$. I have been using ACA_0 as a proxy for this when talking to logicians, but it seems that it might be too strong. – Mario Carneiro Nov 18 '19 at 07:05