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This works aims to map the development of the Löwenheim-Skolems the- orem from Ernst Schröder to Thoralf Skolem using original mathematical notation. It describes its consequence in the form of the Skolem paradox and its influence on set theory and associated issues concerning orders of logic. 1

In the present work, we study algorithms for building finite models of sets of first-order axioms with the aim of proposing and implementing a new method, based on translation onto constraint satisfaction problem (CSP). In the theoretical part, we describe the standard MACE-style method, based on translating problems onto SAT, and advanced techniques that improve the effectiveness of this method: clause splitting, term definitions and static symmetry reduction. Next, we propose an alternative method, which translates problems onto CSP in a similar way. In addition, we have newly proposed a static symmetry reduction technique for binary functions. Next, we describe an implementation of the alternative method using a CSP-modelling language MiniZinc and a CSP-solver Gecode. Finally, we compare performance of our model finder against state-of-the-art representatives of standard methods, systems Paradox and Mace4.