Omega-categorical theory

In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is the special case κ =  = ω of κ-categoricity, and omega-categorical theories are also referred to as ω-categorical. The notion is most important for countable first-order theories.

Equivalent conditions for omega-categoricity

Many conditions on a theory are equivalent to the property of omega-categoricity. In 1959 Erwin Engeler, Czesław Ryll-Nardzewski and Lars Svenonius, proved several independently.[1] Despite this, the literature still widely refers to the Ryll-Nardzewski theorem as a name for these conditions. The conditions included with the theorem vary between authors.[2][3]

Given a countable complete first-order theory T with infinite models, the following are equivalent:

Notes

  1. Rami Grossberg, José Iovino and Olivier Lessmann, A primer of simple theories
  2. Hodges, Model Theory, p. 341.
  3. Rothmaler, p. 200.
  4. Cameron (1990) p.30

References


This article is issued from Wikipedia - version of the 4/26/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.