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There’s also an “invariant definability” argument. I’ll sketch it quickly below, and then give an analysis to explain why I think it’s meaningfully different. Embarrassingly I can't find a source for it at the moment; I recall seeing it as a footnote in Kreisel's model-theoretic invariants paper, but it doesn't seem to be there. Multiple authors have written on invariant definability (which this answer is not-so-secretly an advertisement of) so I haven't yet been able to conduct an exhaustive search for the reference, but when I find it I'll update this. Incidentally, this argument was referred to at the beginning of another answer of mine.

Of course, this is also a feature of the many standard computability-theoretic arguments. I think there’s still a difference here - this time a positive one - due to the way the Tarskian argument interacts with the notion of invariant definability. There are a couple ways to frame this - see e.g. the beginning of this article by Moschovakis for some discussion - and I'll use the following:

There’s also an “invariant definability” argument. I’ll sketch it quickly below, and then give an analysis to explain why I think it’s meaningfully different. Embarrassingly I can't find a source for it at the moment; I recall seeing it as a footnote in Kreisel's Model-theoretic invariants paper, but it doesn't seem to be there. Multiple authors have written on invariant definability (which this answer is not-so-secretly an advertisement of) so I haven't yet been able to conduct an exhaustive search for the reference, but when I find it I'll update this. Incidentally, this argument was referred to at the beginning of another answer of mine.

Of course, this is also a feature of the many standard computability-theoretic arguments. I think there’s still a difference here - this time a positive one - due to the way the Tarskian argument interacts with the notion of invariant definability. There are a couple ways to frame this - see e.g. the beginning of the article Abstract Computability and Invariant Definability by Moschovakis for some discussion - and I'll use the following:

##Argument

##Analysis

Definition:

 

• An arithmetic context is a set $\mathfrak{C}$ of models of Robinson's arithmetic $R$.

 
• For an arithmetic context $\mathfrak{C}$, a set $A\subseteq \mathbb{N}$ is $\mathfrak{C}$-invariantly definable if there is some formula $\varphi$ with $\varphi^M\cap\mathbb{N}=A$ for all $M\in\mathfrak{C}$.

Argument

Analysis

Definition:

• An arithmetic context is a set $\mathfrak{C}$ of models of Robinson's arithmetic $R$.

• For an arithmetic context $\mathfrak{C}$, a set $A\subseteq \mathbb{N}$ is $\mathfrak{C}$-invariantly definable if there is some formula $\varphi$ with $\varphi^M\cap\mathbb{N}=A$ for all $M\in\mathfrak{C}$.

Suppose $M\models T$. Then $Th(M)$ is not pseudo-definable in $M$.

The standard proof still works: let $m$ be the Godel number of the formula $\eta(x)\equiv$ "$\theta$ fails on the number of the sentence gotten by plugging $x$ into the formula with number $x$" - appropriately formalized - and consider the sentence $\eta(\underline{m})$ (applying representability appropriately).

$(*)\quad$ Suppose $M\models T$. Then $Th(M)$ is not pseudo-definable in $M$.

The standard proof still works: supposing to the contrary that $\theta$ pseudo-defined $Th(M)$ in $M$, let $m$ be the Godel number of the formula $\eta(x)\equiv$ "$\theta$ fails on the number of the sentence gotten by plugging $x$ into the formula with number $x$" - appropriately formalized - and consider the sentence $\eta(\underline{m})$ (applying representability appropriately).