Deductive Reasoning in First-order Logic 193 In 1930 Skolem proved that the subsystem of Peano arithmetic involving only axioms for multiplication but not addition, now called Skolem arithmetic in his honor, was complete and decidable, unlike the full system of Peano arithmetic (including both addition and multiplication), which was proven to be incomplete by Gödel in 1931 and undecidable by Church and Turing in 1936 The completeness of first-order logic, first published by Gödel in 1930, follows from results Skolem obtained in the early 1920s and published in 1928 He was apparently not aware of that fact at the time however, because the problem of proving completeness of an axiomatic system for first-order logic was only stated explicitly in 1928 in the first edition of Hilbert–Ackermann’s Principles of Mathematical Logic The most influential and widely recognized work of Skolem was his pioneering research in model theory, the study of applications of logic to mathematics The most well-known result associated with him is the Löwenheim–Skolem Theorem, an extension of an earlier work by Löwenheim published in 1920 It states that if a first-order theory, that is, a set of sentences of first-order logic, has a model (a structure that satisfies them all), then it has a countable model This theorem has some consequences that sound paradoxical, giving rise to what became known as the Skolem paradox: assuming that the axiomatic system ZFC of set theory is satisfied in some model, then it must have one with a countable universe, despite the fact that it can prove the existence of uncountable sets Likewise, the first-order theory of the field of reals, which is uncountable, also has a countable model (However, according to some sources, Skolem did not believe in the existence of uncountable sets, so some of the results associated with such sets may have been falsely attributed to him.) Skolem also provided some of the first constructions of non-standard models of arithmetic and set theory Anatoly Ivanovich Maltsev (14.11.1909–7.06.1967) was a Russian–Soviet mathematician, well known for his work in universal algebra and mathematical logic He is recognized as one of the founders, along with Löwenheim and Skolem, of model theory, one of the main branches of mathematical logic which studies the interaction between mathematics and logic Maltsev studied at Moscow University, completed his graduate work there under A N Kolmogorov and obtained his PhD in mathematics in 1941 from the Steklov Institute of Mathematics, with a dissertation on the Structure of isomorphic representable infinite algebras and groups During 1932–1960 Maltsev taught mathematics at the Ivanovo Pedagogical Institute near Moscow In 1960 he moved to Novosibirsk, where he was head of the department of algebra at the Mathematical Institute of the Siberian branch of the academy, as well as head of the chair of algebra and mathematical logic at the University of Novosibirsk Maltsev obtained very important results on decidability and undecidability of the first-order theories of various important classes of groups and other algebraic