Is there a computably axiomatizable first order theory which is not finitely axiom schematizable?

Question

In this question, I asked what an axiom schema was. In one of the answers, I was told that computably axiomatizable is a vastly weaker notion than "finite number of axiom schemas". Is this really true? Can someone exhibit a computably axiomatizable first order theory which can't be axiomatized by a finite number of axioms and axiom schemas?

Answer 1

Excellent question! The following is just a lengthy comment.

I'm the guilty party. I was speaking imprecisely - I only meant to say that on the face of it computable axiomatizability is a vastly weaker property. Certainly there are lots of examples of computably axiomatizable theories which have no obvious reason to be (what I'll call) scheme-finitely axiomatizable, but it's true that this falls well short of what I said and indeed some of those theories are scheme-finitely axiomatizable after all.

For example, consider Robinson's other arithmetic $\mathsf{R}$, which crops up naturally in the context of gauging the scope of the first incompleteness theorem. $\mathsf{R}$ consists of two infinite sets of axioms: each true quantifier-free sentence is included as an axiom, as is for each $n$ the sentence $$\forall x[\underline{n}<x\vee \bigvee_{i\le n}x=\underline{i}].$$ These families of axioms are not schemes, at least not according to the definition given at the linked answer.

Indeed, I suspect $\mathsf{R}$ is not scheme-finitely axiomatizable.

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That said, however, not too far above $\mathsf{R}$ it becomes rather easy to scheme-finitely axiomatize lots of theories which don't, on the face of it, look scheme-finitely axiomatizable.

Consider the following idea. Let $X=(\chi_i)_{i\in\mathbb{N}}$ be some computable sequence of sentences of bounded quantifier complexity - say, $X\subseteq \Sigma_n$. Now consider the scheme $\sigma_X$ whose "$\varphi$-instance" is the sentence $$\mbox{For all $i$, if $\varphi$ defines $\{i\}$ then $\chi_i$ is true}.$$ This is in fact appropriately expressible in the language of arithmetic, a key point being that since the elements of $X$ have bounded quantifier complexity we can get around Tarski's theorem by using a bounded truth predicate $True$, so $\sigma_X$ makes sense as a scheme - and, in conjunction with a sufficiently strong background theory we in fact get that $\sigma$ is equivalent to $X$! In particular this gives us the following:

If $T$ is an computable set of sentences of bounded quantifier complexity, then $T+I\Sigma_1$ is scheme-finitely axiomatizable.

$I\Sigma_1$ here is certainly a poor upper bound, but I'm erring on the side of caution since it's not immediately clear to me how low we can go after all.

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I suspect that quantifier rank appropriately tweaked is ultimately going to be a useful tool here. Say that the quantifier rank of a scheme $\sigma$ is the smallest quantifier rank of any of its instances (equivalently, the quantifier rank of the result of plugging in "$x=x$" or something similarly boring in for the "varying-formula" bit). Then say that the quantifier rank of a finite set of sentences and schemes is the maximum of the quantifier ranks of the individual sentences and schemes in the set. The hope then is that the finiteness of this bound imposes some strong constraints on the behavior of the theory in question, which we can use to get a non-scheme-finitely-axiomatizable computably axiomatizable theory. However, I don't immediately see how to do this here, and the trick above shows that we can sometimes have scheme-finite axiomatizability for "silly" reasons.

Answer 2

If you allow an infinite language, there are very easy examples. For instance, say you have a unary relation $R_n$ for each $n\in\mathbb{N}$, and you have as axioms $$\forall x R_n(x)$$ for each even $n$. This cannot be axiomatized by finitely many axiom schemes, since such schemes could only involve finitely many of the $R_n$, and therefore the resulting theory would have to be unchanged by any permutation of the $R_n$ that fixes those finitely many. But there is such a permutation that swaps an even and odd $R_n$, so that is impossible.

If you restrict to a finite language, then there should be similar examples, but proving they actually are not axiomatizable by finitely many schemes seems difficult. For instance, you could modify the example above to instead have a constant $0$, a unary operation $s$, and a binary relation $R$, and then take as axioms $$\forall x R(s^n(0),x)$$ for each even $n$. I don't know how to prove this cannot be axiomatized by finitely many schemes, but it seems extremely unlikely that it can be. (For that matter, the same applies to the version that includes such axioms for all $n$ rather than just for all even $n$.) Basically, if you write down some reasonable infinite family of axioms, the resulting theory will be computably axiomatizable, but there will typically be no reason to think the resulting theory can be axiomatized by finitely many schemes. As mentioned in Noah's answer there are some exceptions for theories that can encode enough of arithmetic, but if you have some random very weak theory there is no reason to expect any shenanigans like that.