Incompleteness of Robinson Arithmetic Q and Goedel's theorem

Question

I'm trying to understand the Goedel's theorem. My question is to check if the following thoughts are correct.

1. Goedel considers the formal sistem Q which describes natural numbers with addition and multiplication without any induction axioms. If we add a first order induction scheme to Q we obtain first order Peano PA. The key point, for me to understand, is that Q and PA share the same formulas, so any proposition in PA can also be written in Q. Of course in PA we have more theorems than in Q so proving that Q is incomplete does not mean that also PA is incomplete.

2. For example I believe that the formula "given any number n there exists a prime number p greater than n" can be expressed in Q is not provable in Q but, is provable in PA.

3. In the first theorem Goedel finds a formula G in Q which can be interpreted as "G is not provable in PA". Actually for any formal system he is able to find a formula in Q which is not provable in that formal system (if we mean a system where each theorem can be enumerated by means of a Turing machine). For example: ZF is also incomplete.

4. If we are only interested in ZF we can slightly simplify Goedel's proof avoding the introduction of the $\beta$-function. In fact the $\beta$-function is needed to prove that any recursive function can be represented in Q but it is clear that any recursive function can be represented in ZF.

Answer 1

(1) is mostly right. It's wrong, though, to think of Godel's argument as honing in on Q specifically. First of all, Godel originally looked at Russell and Whitehead's theory PM, not Q or PA. More to the point, Godel's argument (+ a later improvement by Rosser) "immediately relativizes upwards" - if $T$ is a theory for which Godel's argument goes through, then in fact we get immediately that any recursively axiomatizable extension of $T$ is also incomplete.

Q comes into play when we try to get an optimal form of Godel's theorem - that is, we look for an as-weak-as-possible theory $T$ such that Godel's argument works for $T$, and Q serves well as such a theory. The point, though, is that - while incompleteness of Q doesn't imply the incompleteness of PA outright - the argument for Q's incompleteness immediately lifts to PA. A pretty-optimal form of GIT is the following:

Suppose $T$ is a recursively axiomatizable consistent theory which interprets Q. Then $T$ is incomplete.

(Here "interprets" is a rather technical property; for now, you can read it as "contains." But it is important that it's broader than mere containment - for instance, ZFC doesn't literally contain Q since $\{+,\times, 0,1\}$ isn't a sublanguage of $\{\in\}$, but ZFC does interpret Q.)

---

(2) is right. See also this old MSE question. I think the snappiest example of an "obviously true" sentence not provable in Q is "Every number is either even or odd" (formally: $\forall x\exists y(y+y=x\vee y+y+1=x)$).

---

For (3), we have to be careful about the formal theories we have in mind. There are plenty of complete theories containing PA (e.g. the full theory of the structure $(\mathbb{N};+,\times,0,1)$), and recursively axiomatizable complete theories (e.g. the full theory fo the structure $(\mathbb{R}; +,\times,0,1)$; perhaps surprisingly, this is much simpler than $\mathbb{N}$!). See my section on (1) for a careful way to phrase GIT.

---

For (4), I would disagree with the larger point. ZF is a theory just about $\in$. We have to build a "theory of finite sequences" in the language of set theory, and that's in my opinion as tedious as, if not more than, the situation for the language of arithmetic. ZF does remove the need for the $\beta$ function, but it introduces other difficulties.

If you truly want to make things easier, the right thing to do is add exponentiation to arithmetic so you can code sequences via prime factorization. When I present GIT, this is how I do it: I show that PA$_{exp}$ is incomplete (if it's consistent of course), where PA$_{exp}$ is the analogue of PA for the larger languaeg $\{+,\times,0,1,exp\}$.