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

Question

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?

Answer 1

It seems like you're a little confused about the way Gödel's proof works. Let me try to clarify this, which I hope will shed some light on the whole question:

• 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!