# How can we know whether an arbitrary sentence corresponds to a sentence in the language of arithmetic or not?

From what I came to understand from Godel's work is that a consistent effectively generated theory $$T$$ can have its consistency statement $$Con(T)$$ written in the language of $$T$$ itself! and that this wouldn't be provable in $$T$$ should $$T$$ be strong enough to formulate basic arithmetic. Now also I came to understand from this answer, that this statement being in the language of $$T$$ itself would be coded as a "natural number" which is of course the Godel number for that sentence. But at the same time there is an arithmetical sentence $$Con(T)^{arith}$$ that is TRUE in the standard model $$\mathbb N$$ of arithmetic! So we have $$Con(T)$$ being equivalent to the syntactical meta-theoretic statement $$T \text{ is consistent}"$$ and also we have $$Con(T)^{arith}$$ being equivalent to that meta-theoretic consistency statement. This is a form of correspondence whereby a sentence is the language of $$T$$ corresponds to a sentence in the language of arithmetic, and not just to a term in that language.

Question: is there a general way to determine whether an arbitrarily given sentence $$\phi$$ in the language of $$T$$ corresponds to a sentence in the language of arithmetic or just correspond to some natural number?

• If $$T$$ is an effectively generated first-order theory, then there is a sentence in the language of arithmetic, $$\text{Con}(T)^{\text{arith}}$$ (adopting your notation), expressing the consistency of $$T$$.
• If $$T$$ is strong enough to formulate basic arithmetic - and let's be precise about this by assuming $$T$$ interprets PA, though this is stronger than what's actually required - then $$\text{Con}(T)^{\text{arith}}$$ can be translated into a sentence $$\text{Con}(T)$$ in the language of $$T$$.
• If $$T$$ is consistent, then $$T$$ cannot prove $$\text{Con}(T)$$.
It's the second point that's most relevant to your question. The sense in which $$\text{Con}(T)$$ (as a sentence in the language of $$T$$) corresponds to a sentence in the language of arithmetic is no more or less than the fact that it's the translation of $$\text{Con}(T)^{\text{arith}}$$ via the interpretation of PA in $$T$$.
Taking this as the meaning of "corresponds to a sentence in the language of arithmetic", the answer to your question is yes: the question of whether a sentence of the language of $$T$$ corresponds to a sentence in the language of arithmetic is algorithmically decidable.
But it's really not very interesting. For most set theories $$T$$, the interpretation of PA in $$T$$ will amount to taking the domain to be $$\omega$$ and the arithmetic operations to be the usual ordinal arithmetic operations on $$\omega$$. Then a sentence in the language of $$T$$ is in the image of this interpretation just when it is built up from equations involving only the ordinal arithmetic operations, using Boolean connectives and quantifiers that are explicitly bounded to $$\omega$$. That is, if it obviously expresses a (first-order in the language of arithmetic) property of the natural numbers!