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A function $f:\mathbb{N}^k\rightarrow\mathbb{N}$ is arithmetic iff its graph is arithmetic, i.e., there is a formula $\psi(\vec{x},y)$ in the language of Peano arithmetic such that for all $\vec{a}$ in $\mathbb{N}^k$, $f(\vec{a})=b$ iff $\mathbb{N}\models\psi(\vec{a},b)$.

For $f$ to be represented in PA, we have the (much) stronger condition that for all $\vec{a}$ in $\mathbb{N}^k$, $f(\vec{a})=b$ iff $\text{PA}\vdash\psi(\vec{a},b)$.

I'm wondering about an intermediate condition: $f$ is arithmetic, and in addition, PA proves that the associated formula defines a function. That is, $\text{PA}\vdash\forall\vec{x}\,\exists!y\,\psi(\vec{x},y)$.

Would it be reasonable to call this condition expressible in PA? Or would that conflict with generally accepted terminology?

I'm also assuming that definable in PA is a synonym for arithmetic. But I'm interested in hearing from anyone who doesn't agree with this.

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  • $\begingroup$ IMO, the terminology is not very "stable". R.Smullyan (into his G's Incompl Th) uses expressible for the first one (true in...) and representable for the second one (provable in...). $\endgroup$ – Mauro ALLEGRANZA May 10 at 14:59
  • $\begingroup$ Your condition for "represented" doesn't make sense : what's the connection between $a$ and $x$, $b$ and $y$ ? $\endgroup$ – Max May 10 at 15:39
  • $\begingroup$ @Max I assume what was meant was $\text{PA}\vdash \psi(\overline{a},b)$. $\endgroup$ – Alex Kruckman May 10 at 15:52
  • $\begingroup$ @AlexKruckman yes I thought so but at first I had a misconception in my head about what $\mathbb{N}\models \psi(a,b)$ implied about provability in PA, I was thinking in too low complexity of formulas $\endgroup$ – Max May 10 at 16:00
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Your condition is called definable. This is standard usage in model theory: Given a theory $T$, a definable function is a formula $\varphi(\overline{x},y)$ such that $T\vdash \forall \overline{x}\, \exists^! y\, \varphi(\overline{x},y)$, up to provable equivalence in $T$. Then for any model $M\models T$, $\{(\overline{a},b)\in M^{n+1}\mid M\models \varphi(\overline{a},b)\}$ is the graph of a function $M^n\to M$ (which we abuse terminology by also calling a definable function).

It would be a big mistake to make "definable in PA" a synonym for "arithmetic". The point is that the definition of arithmetic has nothing to do with the theory PA, it's just about the standard model $\mathbb{N}$. On the other hand, the definition of definable has nothing to do with the standard model $\mathbb{N}$, it's just about the theory PA. These are totally different concepts.

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  • $\begingroup$ OK, thanks. And I guess we would use "definably represented" for the conjunction of those two conditions. (A priori, I can't see why one would imply the other, in general (i.e., for $\psi$ high enough in the hierarchy). $\endgroup$ – Michael Weiss May 10 at 18:39
  • $\begingroup$ @MichaelWeiss Sure, that sounds like reasonable terminology. I agree that it doesn't seem like definable and represented are comparable notions, though I don't have counterexamples in mind). $\endgroup$ – Alex Kruckman May 10 at 18:44

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