Omega-categorical theory

The set of Ryll - Nardzewski is a set of model theory, a branch of mathematical logic. He characterizes categorical theories. It is named after the Polish mathematician Czesław Ryll - Nardzewski.

Set of Ryll - Nardzewski

Be a complete theory over a countable language. With the space of complete types is called.

Then is equivalent to:

  • Is - categorical.
  • For all finite.
  • Up to equivalence, there are only finitely many formulas for each

Other equivalences

Under the same assumptions as in the theorem of Ryll - Nardzewski applies that is equivalent to:

  • Is - categorical.
  • Every countable models is saturated.

Examples

Density linear order without endpoints

Be a model of the theory of dense linear order without endpoints and

And without limiting the generality

A complete type over either by a formula of the form:

Or Form

Generated. This can be proved by quantifier elimination.

The set of types is finite, the theory is so - categorical

Theory with infinite many constant symbols

The theory of language with the axioms has countably many complete 1- types: The types generated by the formula, the isolated types of generated by the set type is the only non-isolated type. The theory is therefore non- categorical. ( It is, however, categorically. )

Saturated model Wilfrid Hodges
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