# Why is $\{\#F\mid\mathcal{N}\models F \text{ and$F$is an$\exists$-formula} \}$ a recursive set?

Let $$\mathcal{N}=(\mathbb{N},+,\cdot,0,1)$$. I want to show that $$\{\#F\mid\mathcal{N}\models F \text{ and F is an \exists-formula} \}$$ is a recursive set. Here $$\#F$$ is the Gödel number of $$F$$ and I think that $$\exists$$-formula just means a formula on the form $$\exists v_0 \dots\exists v_n F$$, without any occurrence of universal quantifiers. The problem is from an old mock-exam from a second-level logic course given at my university.

This was my original attempt, which, as Carl Mummert pointed out, is clearly wrong:

My idea is to show that $$\{\#F\mid\mathcal{N}\models F \}$$ is recursive and then just multiply its characteristic funtion with something like $$f(x)=\left[1\,\dot{-}\,sg(\vert \beta_3^3(x)-7\vert )\right]$$, i.e. something that takes the Gödel number of $$x$$ and returns $$0$$ if the last digit is not $$7$$ and $$1$$ if it is (i.e. if it is an $$\exists$$-formula). That would give me the characteristic funtion for the original set. But then my problem is that I don't know how to show that $$\{\#F\mid\mathcal{N}\models F \}$$ is recursive.

• The set of true formulas of arithmetic - $\{ \#\phi : \mathbb{N} \vDash \phi\}$ - is very far from being recursive. So that method will not work. Instead you would need to look at the special structure of a $\exists$ formula and use that to show that there is a way specific to those formula to tell if they are true. – Carl Mummert Jan 15 at 16:58
• You could improve the post by adding a little more background. What book is the problem from? What is your specific definition of a $\exists$ formula? – Carl Mummert Jan 15 at 16:59
• Yes of course, now that you phrase it that way... The problem is from an old mock-exam from a logic course given at my university. Now, unfortunately this is the first time I see the term "$\exists$-formula" but I am assuming they just mean that it is on the form $\exists v_0 G$. But thanks, I'll add that to my question. – KurtKnödel Jan 15 at 17:07
• \mid produces the correct spacing. – Asaf Karagila Jan 15 at 18:09
• Are you sure the text you got the problem from intended you to show the set is recursively decidable, rather than recursively enumerable (which is semidecidable) ? – DanielV Jan 15 at 21:10

## 1 Answer

It's not recursive.

Whether this is easy or very hard to show depends on exactly what "$$\exists$$-sentence" means. Specifically, the issue is whether "bounded quantifiers" (like $$\forall x<5$$, $$\exists y) are permitted. Formulas of the form $$\exists x_1,...,x_n G$$ where $$G$$ uses only bounded quantifiers are called "$$\Sigma_1$$" (or "$$\Sigma^0_1$$"); to me, "$$\exists$$-formula" sounds like a stronger condition, namely we require that the matrix $$G$$ be genuinely quantifier-free (and this is your interpretation in the comments above).

But in each case, the resulting set is non-recursive.

Showing that the $$\Sigma_1$$ theory of $$\mathcal{N}$$ is non-recursive isn't too hard. There is a $$\Sigma_1$$ formula $$Halt(x)$$ such that for each $$n\in\mathbb{N}$$ we have $$\mathcal{N}\models Halt(\underline{n})$$ iff $$\varphi_n(n)\downarrow$$ - writing "$$\varphi_e(x)\downarrow$$" for "the $$e$$th Turing machine halts on input $$x$$" and "$$\underline{k}$$" for the numeral corresponding to $$k$$ - but this means that the $$\Sigma_1$$-theory of $$\mathcal{N}$$ is at least as complicated as (in fact, it's equivalent to) the halting problem, which is non-recursive.

Gauging the complexity of the genuinely $$\exists$$-theory of $$\mathcal{N}$$ is harder, but ultimately still the same. This is the MRDP theorem: that the set of Diophantine equations with solutions (which is about as $$\exists$$ as it gets) is non-recursive, and in fact has complexity again equal to that of the halting problem.

• Thank you very much for this! This explains why it's been so hard to prove it... Actually, the original problem was to show that $\{F\,\vert\,\mathcal{N}\models F \text{ and$F$is an$\exists$-formula} \}\cup PA$ is an incomplete theory. Do you reckon this is still possible (using the first incompleteness theorem), despite the theory not being recursive? – KurtKnödel Jan 15 at 17:56
• @KurtKnödel Ah, that's a very different problem. It is indeed correct, and I have two hints. $(1)$ For a "nuke," use Craig's trick and the strong version of GIT ("no consistent recursively axiomatizable extensionof PA is complete"). $(2)$ For the optimal solution, show that PA already proves every true $\exists$-sentence, so you don't even need strong GIT. – Noah Schweber Jan 15 at 17:58
• Your second suggestion was actually my idea all along, but then somewhere someone mentioned recursive sets and I just lost it at that point. Thank you for clarifying things! – KurtKnödel Jan 15 at 18:06