In (first-order) logic, I understand that there are two notions of the truth of a sentence $\phi$ in a theory $T$:
- Syntactic truth: $T\vdash \phi$ if $\phi$ is provable from $T$,
- Semantic truth: $T\models \phi$ if $\phi$ is true in every model $M$ of $T$.
Naturally, both the notion that "$\phi$ is provable from $T$" and that "$\phi$ is true in every model $M$ of $T$" must be formulated in some metatheory $T'$, and the details of this metatheory will determine the truth of these assertions. We might formulate $T$ in a metatheory which shows that $T\models \phi$ for some sentence $\phi$, or we might formulate it in a metatheory which shows $\neg (T\models \phi)$, or we might formulate it in a metatheory which does not show either one. So the truth of $\phi$ in $T$ seems to depend in an essential way on the metatheory.
Am I correct so far in my understanding?
Supposing that I am, here is where my confusion lies. If the truth of $\phi$ in $T$ depends on the metatheory $T'$, then in order to determine if $\phi$ is syntactically true in $T$ we must consider the truth of the sentence $T\vdash \phi$ in $T'$, and in order to determine if $\phi$ is semantically true in $T$ we must consider the truth of the sentence $T\models \phi$ in $T'$. In both cases we must regress to a new metatheory $T''$, where we are now concerned with the truth values of all of
- $T' \vdash (T\vdash \phi)$
- $T' \models (T\vdash \phi)$
- $T' \vdash (T\models \phi)$
- $T' \models (T\models \phi)$
in $T''$.
In general, the number of notions of truth we may consider grows as $2^n$ with the number of metatheories $n$ we invoke. Each of these notions of truth can be true, false, or unprovable in the metatheory. Now, clearly this is a naive upper bound on the hairiness of the situation: soundness rules out many scenarios, such as $(T\vdash \phi) \land \neg (T\models \phi)$. First-order logic has the completeness theorem, which likewise rules out $\neg (T\vdash \phi) \land (T\models \phi)$, although higher-order logic does not. I assume other metamathematical observations may pare things down as well. Of course, the truth of these metamathematical facts will again depend on the metatheory, so the situation still seems somewhat hairy. How is this dealt with?
More fundamentally, if truth must be defined with reference to a metalanguage, how do we avoid infinite regress? I gather that in practice the answer is something like this: we fix an outermost metatheory $T_{fin}$ which remains informal, and for which we do not attempt to explicitly define any notion of truth. Rather, we take truth in the informal $T_{fin}$ as a primitive, and trust that we "know it when we see it". This may be aided by using an outermost theory $T_{fin}$ which does not invoke any particularly intuitively vexing concepts, such as uncountable infinities. By invoking $T_{fin}$, we end the infinite regress, because truth of a sentence $\phi$ in $T_{fin}$ remains formally undefined and so does not involve regress to another metatheory.
Is this understanding accurate?
If so, that is somewhat philosophically vexatious, although it does not seem there is much that can be done about it.
Edit: this question has been marked as a duplicate of "Are there infinitely many metalogics?". While I agree that the basic thrust of that question is the same as my own, I believe that my specific inquiry about exponential proliferation of "truth notions" in an $n$th-level meta language is not asked about there or addressed in any of the answers. I am also not sure if it specifically addresses my question about (what I called above) $T_{fin}$, and if my account is a correct understanding of how we are implicitly defining truth at the outermost level when we prove metalogical results.
