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<p>We investigate the expressive power of first-order logic based on two levels: finitary and infinitary. A complex of unifying transformations of theories is introduced, including signature reduction and the universal construction of finitely axiomatizable theories. For each level, the corresponding layers of model-theoretic properties are defined, and globalization formulas describing the Tarski–Lindenbaum algebra of predicate calculus of a finite rich signature are established. It is shown that the finitary layer covers all model-theoretic properties, while the infinitary layer corresponds to properties preserved under reduction of infinite signatures to finite ones. The results obtained generalize and refine Tarski’s classical problem on the characterization of the Lindenbaum algebra.</p>

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Two Levels of Expressiveness of First-order Logic

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • M. G. Peretyatkin

摘要

We investigate the expressive power of first-order logic based on two levels: finitary and infinitary. A complex of unifying transformations of theories is introduced, including signature reduction and the universal construction of finitely axiomatizable theories. For each level, the corresponding layers of model-theoretic properties are defined, and globalization formulas describing the Tarski–Lindenbaum algebra of predicate calculus of a finite rich signature are established. It is shown that the finitary layer covers all model-theoretic properties, while the infinitary layer corresponds to properties preserved under reduction of infinite signatures to finite ones. The results obtained generalize and refine Tarski’s classical problem on the characterization of the Lindenbaum algebra.