14 Facts About Model theory

In mathematical logic, model theory is the study of the relationship between formal theories, and their models.

Relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within a model fluctuated in the history of the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively:.

For instance, while stability was originally introduced to classify theories by their numbers of models in a given cardinality, stability theory proved crucial to understanding the geometry of definable sets.

The corresponding notion in model theory is that of a reduct of a structure to a subset of the original signature.

Similarly, if s' is a signature that extends another signature s, then a complete s'-Model theory can be restricted to s by intersecting the set of its sentences with the set of s-formulas.

The corresponding concept at the level of theories is called strong minimality:A theory T is called strongly minimal if every model of T is minimal.

In particular, any ultraproduct of models of a theory is itself a model of that theory, and thus if two models have isomorphic ultrapowers, they are elementarily equivalent.

The Lowenheim–Skolem theorem implies that if a theory T has an infinite model for some infinite cardinal number, then it has a model of size ? for any sufficiently large cardinal number ?.

Fundamental result in stability Model theory is the stability spectrum theorem, which implies that every complete Model theory T in a countable signature falls in one of the following classes:.

Furthermore, if a Model theory is -stable, it is stable in every infinite cardinal, so -stability is stronger than superstability.

Many construction in model theory are easier when restricted to stable theories; for instance, every model of a stable theory has a saturated elementary extension, regardless of whether the generalised continuum hypothesis is true.

The first significant result in what is model theory was a special case of the downward Lowenheim–Skolem theorem, published by Leopold Lowenheim in 1915.

An example of an influential proof from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields.

Finally, some questions arising from model theory have been shown to be equivalent to large cardinal axioms.