# Can we have an axiom that refers to itself and the prior axioms of the theory it is an axiom of?

I know that this question is little bit imprecise, I'll try to present it in the best I can.

Can one have an axiom which is self referential with respect to itself and the theory in which it belongs?

In particular I'm concerned with an axiom which says that: there exists an $$\omega$$-model of the theory axiomatized by all axioms of this theory except this axiom.

So, we have a theory $$\mathsf T$$, and among the axioms of $$\mathsf T$$, there is an axiom $$\psi$$ stating that there exists an $$\omega$$-model of $$\mathsf T-\psi$$. Or for example if there is a way to add a qualification of the axioms prior to $$\psi$$, call it "prior axiom" within the language of $$\mathsf T$$ and state $$\psi$$ as existence of $$\omega$$-model of the theory axiomatized by all sentences labeled "prior axiom".

The general context of this question is that I intend to use it in proofs by compactness, so we prove that every finite prior axioms and $$\psi$$ is consistent, then the theory made from the infinitely many all prior axioms and $$\psi$$ would be consistent by compactness, but here it'll have an $$\omega$$-model for the prior fragment of it.

Can this be done for some theories, how, and for which theories can it be done?

• You can have robust self-referentiality making assertions about syntactic features of the prior axioms and itself and the theory. So consistency is fine, but not generally about the truth or falsity of those axioms in other contexts, since by Tarski's theorem truth is generally not expressible. Apr 9 at 21:21
• First determine what $S:=T-\varphi$ should be, and then just let $\varphi$ be "$S$ has an $\omega$-model" (or similar). There's no self-reference here. Self-reference would occur if you wanted $S+\varphi$ to say "$S+\varphi$ has a model" (or similar), but that's not what you're doing here. Apr 9 at 21:47
• @NoahSchweber, My plan is to use that feature in an argument by compactness, so that the final result would be a theory having an axiom asserting the existence of $\omega$-model the rest of it. So in reality $T$ would range over several theories (finite axiomatizations) that have $\psi$ as an axiom, so here you have $\psi$ saying that it is an axiom of $T$ and there is an $\omega$-model of the fragment of $T$ axiomatized by the axioms of $T$ other than $\psi$. So, actually there is double self reference here of being an axiom of a theory and speaking of the rest of the theory. Apr 10 at 6:49
• @JoelDavidHamkins, so per your comment, I understand that this can be done! Since I'm referring to a special context in reference to the theory in which the statement is an axiom. Apr 10 at 6:58

For example, one can consider the provability version of the Visser-Yablo paradox, a Gödelized infinite liar, which produces a uniformly expressible sequence of sentences $$\sigma(k)$$ in the language of arithmetic, each of which provably expresses the non-provability of all the later sentences $$\text{PA}\vdash\quad\forall k\left[\sigma(k)\leftrightarrow\forall n{>}k\ \left(\strut\neg\text{Pr}_{\text{PA}}(\ulcorner\sigma(\bar n)\urcorner)\right)\right].$$ I consider these sentences in my elementary essay on the Infinite Liars. So, the main lesson is that self-reference is no problem. With quite flexible methods, one can find a solution to nearly any self-referential expressible condition.
Let me emphasize, however, that self-referentiality means that sentences can refer to any expressible properties of themeselves and the theory in which they are described. This would include any kind of syntactic feature of the sentences, such as provability, consistency, satisfiability, but also $$\omega$$-realizability. In set theory one can form self-referential theories that refer to the $$\omega$$-realizability of themselves or other parts of the sentences.
In particular, we can form form a formula $$\varphi(x)$$ such that ZFC proves that for every natural number $$k$$, $$\varphi(k)$$ asserts that the theory arising from $$\varphi(n)$$ for $$n\neq k$$ is true in some $$\omega$$-standard model. The existence of such a formula is an immediate instance of the Gödel-Carnap fixed point lemma.
If we would mean "an $$\omega$$-model of ZFC" for the sentences, then it is consistent with ZFC that they are false, since it is consistent with ZFC that there is no $$\omega$$-model of ZFC at all. I'd have to think more carefully to determine if they can be true, or what happens if one wants a model just of this theory and not also the ambient theory, such as ZFC here.