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Does there exist a theory $T$ (in the sense of model theory) such that $T$ is recursively axiomatizable, but there is no independent recursive axiomatization of $T$? Or, does every recursively axiomatizable theory have an independent recursive axiomatization? By independent axiomatization, I mean an axiomatization where no sentence is redundant.

user107952
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  • If T is recursively axiomatizable, I can enumerate its axioms $\varphi_0,\varphi_1,\dots$, so I could not use the same approach as the proof that every class of structures $\Delta$-elementary has an independent set of axioms? Getting $\Phi_0=\emptyset$, $\Phi_i=\Phi_{i-1}$ if $\varphi_1,\dots,\varphi_{i-1}\vDash\varphi_i$ and $\Phi_{i-1}\cup{\varphi_1\wedge\dots\wedge\varphi_{i-1}\to\varphi_i}$ otherwise? – Xennonio Dec 25 '23 at 14:36
  • @Xenônio That's highly non-recursive: how do you tell whether $\varphi_1,...,\varphi_{i-1}\models\varphi$? – Noah Schweber Dec 27 '23 at 21:50
  • oh, that's right @NoahSchweber, this just proves that there is an independent axiomatization, my bad, thank you for the explanation – Xennonio Dec 27 '23 at 22:32
  • I think Kreisel came up with an example of what you are looking for. See https://www.jstor.org/stable/2964108?seq=5. – Rob Arthan Jan 12 '24 at 23:00
  • @RobArthan That should probably be an answer. – user107952 Jan 13 '24 at 03:36

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As pointed out by Rob Arthan in a comment above, this was resolved by Kreisel in a 1956 JSL abstract. The picture was further clarified by Pour-El, Independent axiomatization and its relation to the hypersimple set in 1968, who proved the following theorem:

Let $T$ be a first-order theory in a recursive language. Then the following are equivalent:

  • $T$ has no recursive independent axiomatization.

  • For some axiomatization $\{\alpha_i:i\in\omega\}$ of $T$, the set $\{i: \bigwedge_{j<i}\alpha_j\vdash \alpha_i\}$ is hypersimple.

  • For every axiomatization $\{\alpha_i:i\in\omega\}$ of $T$, the set $\{i: \bigwedge_{j<i}\alpha_j\vdash \alpha_i\}$ is hypersimple.

This answered a question implicitly raised at the end of Kreisel's abstract, namely whether there was a real connection between non-recursive independent axiomatizability and hypersimplicity (Kreisel merely observed that in his example one could not replace hypersimplicity with simplicity).

A natural follow-up question is whether an analogue of Pour-El's theorem (or even of Kreisel's counterexample) exists for $\mathcal{L}_{\omega_1,\omega}$ - presumably working under $\mathsf{V=L}$ and using the theory of $\omega_1$-recursion. As far as I know this has not been explored, but X. Caicedo's result that size-$\aleph_1$ $\mathcal{L}_{\omega_1,\omega}$-theories do have independent axiomatizations (see e.g. these slides of Souldatos) sets the stage nicely.

Note that Pour-El uses the slightly more ambiguous term "independent axiomatization" in place of "recursive independent axiomatization;" to a modern eye, her claim may appear in contradiction with the (easy) result that every countable first-order theory does have an independent axiomatization. (In older texts especially, "axiomatization" generally means "recursive axiomatization," which personally I dislike.)

Noah Schweber
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