The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, o-minimal structures, to name a few. We give an overview of the algebraic and model-theoretic aspects of this differential field, and report on our efforts to understand its first-order theory.

We generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results relating involutivity, restrictions, characteristics and characteristicity, known for first order systems, extend to the general context. This involves, in particular, a new definition of strong characteristicity. The proof exploits a spectral sequence relating S...

In this paper, we characterize the logical consequence relation as the minimal relation in a class of 'adequate consequence relations for quantification theory. These adequate relations are easily seen to include the usual syntactic relations of deductive consequence and thus our results can, perhaps, be considered an abstract version of the completeness theorem. Our work is closely related to ...

A first-order sentence θ of vocabulary σ ∪ {S} is successor-invariant in the finite if for every finite σ-structure M and successor relations S1 and S2 on M, (M, S1) |= θ ⇐⇒ (M, S2) |= θ. In this paper I give an example of a non-first-order definable class of finite structures which is, however, defined by a successor-invariant first-order sentence. This strengthens a corresponding result for o...

In this paper we define computationally well-behaved versions of classical first-order logic and prove that the validity problem is decidable1.

I defend the Received View on scientific theories as developed by Carnap, Hempel, and Feigl against a number of criticisms based on misconceptions. First, I dispute the claim that the Received View demands axiomatizations in first order logic, and the further claim that these axiomatizations must include axioms for the mathematics used in the scientific theories. Next, I contend that models are...

We consider successor-invariant first-order logic (FO+ succ)inv, consisting of sentences Φ involving an “auxiliary” binary relation S such that (A, S1) |= Φ ⇐⇒ (A, S2) |= Φ for all finite structures A and successor relations S1, S2 on A. A successorinvariant sentence Φ has a well-defined semantics on finite structures A with no given successor relation: one simply evaluates Φ on (A, S) for an a...