On Triggers of Order
DOI:
https://doi.org/10.31489/2026m1/82-86Keywords:
proper heir, proper coheir, strong heir, unstable theory, non-definable type, order property, trigger of order, ω-evidenceAbstract
It is shown that the concepts of heir and coheir, introduced by D. Lascar and B. Poizat, play a fundamental role in model theory, particularly in classification theory. The related notions of proper heir and proper coheir are introduced, containing important constructs within themselves. Poizat’s lemma on the existence of a proper heir of any non-definable type over a model is presented as an important fact of existence in unstable theories. The concept of an order trigger in a model is then introduced as the skeleton of an algorithmic device that produces ω-evidence of the order property in it. This evidence is constructed using a method very similar to the “back and forth” method of classical model theory, where at each step two possibilities for choosing elements are alternated. As an example of use, a simplified proof of the characterization theorem of the class of unstable theories using these concepts is explained. It is pointed out that applications of more advanced constructions, such as order triggers, can help in solving problems related to the classification of small, countable and minimal models of unstable theories.
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