Article · BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS · June 26, 2026
Fractal completeness and compactness in first-order model theory
Aibat R. Yeshkeyev, N.V. Popova, A.K. Issayeva, M.T. Kassymetova
Abstract
This article develops a formal framework for fractal structures within classical first-order model theory. The notion of fractality is reformulated in purely logical terms by replacing metric self-similarity with logical self-similarity induced by elementary endomorphisms of structures. A hierarchy of morphisms is introduced, including endomorphisms, elementary endomorphisms, and endiks, which preserve the truth of formulas in one or both directions. Based on these morphisms, fractal subsets and fractal models are defined via finite families of elementary self-maps. On the syntactic level, fractality is expressed through finite systems of T-elementary syntactic endomorphisms generating a stabilization process called the fractal corridor (a sequence of theories generated by iterated application of syntactic endomorphisms). A compatibility condition between syntactic and semantic fractality is formulated and proved. Using a Henkin-type construction, syntactic operators are lifted to semantic self-maps of a canonical model, yielding fractal completeness. A corresponding compactness theorem is also established. All constructions are carried out within standard first-order logic.
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