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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? – Xenônio 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 – Xenônio 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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