Pseudofiniteness in Hrushovski Constructions

Notre Dame Journal of Formal Logic 61 (1):1-10 (2020)
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Abstract

In a relational language consisting of a single relation R, we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation R plays a crucial role in this context. When R is ternary, by extending the methods recently developed by Brody and Laskowski, we interpret 〈Q+,<〉 in the 〈K+,≤∗〉-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in an earlier work by Evans and Wong. This result, in fact, unfolds another aspect of complexity of this structure, along with undecidability and the strict order property proved in the mentioned earlier works. On the other hand, when R is binary, it can be shown that the 〈K+,≤∗〉-generic is decidable and pseudofinite.

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Stable generic structures.John T. Baldwin & Niandong Shi - 1996 - Annals of Pure and Applied Logic 79 (1):1-35.
On generic structures.D. W. Kueker & M. C. Laskowski - 1992 - Notre Dame Journal of Formal Logic 33 (2):175-183.
Simple generic structures.Massoud Pourmahdian - 2003 - Annals of Pure and Applied Logic 121 (2-3):227-260.
On rational limits of Shelah–Spencer graphs.Justin Brody & M. C. Laskowski - 2012 - Journal of Symbolic Logic 77 (2):580-592.
Smooth classes without AC and Robinson theories.Massoud Pourmahdian - 2002 - Journal of Symbolic Logic 67 (4):1274-1294.

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