The above applies to first order theories, such as Peano Arithmetic.
2.
Sela also proved that the first order theory of a finitely generated free group is stability theory.
3.
Following that he has extended the work far beyond first order theories, for example for abstract elementary classes.
4.
No axiom requires more than three quantified variables; hence a translation of first order theories into relation algebra is possible.
5.
Given a complete first order theory T, the set of structures satisfying T is a minimal, closed invariant set, and hence Polish in its own right.
6.
In particular, paper ( 2 ) contains what is now called " Svenonius'Theorem ", an important result on definability of predicates in first order theories.
7.
However, for a specific model that may be described by a first order theory, some statements may be true but undecidable in the theory used to describe the model.
8.
This is due to Alfred Tarski . ) The first order theory of the natural numbers with addition ( but not multiplication ), called Presburger arithmetic, is also decidable.
9.
A "'model "'of a first order theory \ mathcal { QS } is an interpretation in which every formula of \ mathcal { QS } is satisfied.
10.
Despite this theoretical limit, in practice, theorem provers can solve many hard problems, even in models that are not fully described by any first order theory ( such as the integers ).
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