Theory of finite sets of naturals (with order)

Question

Consider the two-sorted structure consisting of the natural numbers with the usual ordering together with the set of finite sets of naturals equipped with "$\in$."

I recall seeing a proof that this structure is decidable (that is, its full elementary diagram is computable) - indeed, that the same is true if we include addition - but I can't find it now. I have a sketch of an argument, but it's tedious, and I'd rather have a citation than fill in the details (especially if I'mmisremembering!).

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It's worth noting that there are some things we can define in this structure. For example, the set of even numbers can be defined as follows: $n$ is even iff there are finite sets $A,B$ such that

• $A\cup B$ is closed downwards,

• $0,n\in B$,

• for all $b\in B$ we have $b+1\in A$, and

• for all $a\in A$, if $a<n$ then $a+1\in B$.

(Similarly, each residue class is definable.) So it's not entirely boring, model-theoretically.

Answer 1

The logic you're referring to is called WMSO[<], or Weak Monadic Second Order Logic (with <).

It is the logic where one is allowed to quantify over (monadic) second order terms (i.e., the subsets of $\mathbb{N}$), but only weakly (i.e. only the finite subsets).

There are many results known about this and related logics, as Buchi found a kind of automataon which allows decidability results to be found somewhat naturally. The only proof of this theorem I'm familiar with comes from the theory of Buchi Automata. It can be found in:

"Weak Second-Order Arithmetic and Finite Automata" - Buchi

Unfortunately, Buchi's notation is horrid, and you might have better luck reading:

"Decision Problems of Finite Automata Design and Related Arithmetics" - Elgot

Normally I would sketch the proof here for ease of reference, but unfortunately it's rather gross... The idea is to find a normal form for formulas in this logic, then show that these normal forms correspond to infinite strings over an alphabet, which we can feed as input to Buchi Automata. One then shows how to decide the accepting language of Buchi Automata, and after showing that some automaton exists which accepts exactly the "true" formulas, the proof is complete.

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I hope this helps! ^_^