This conjecture states that there is an effective procedure that, given n ≥ 1 and exponential polynomials in n variables with integer coefficients f1,..., fn, g, produces an integer η ≥ 1 that depends on n, f1,..., fn, g, and such that if α ∈ Rn is a non-singular solution of the system
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As Stan Wagon points out at the end of his monograph, the Banach–Tarski paradox has been more significant for its role in pure mathematics than for foundational questions: it motivated a fruitful new direction for research, the amenability of groups, which has nothing to do with the foundational questions.
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Banach and Tarski explicitly acknowledge Giuseppe Vitali's 1905 construction of the set bearing his name, Hausdorff's paradox (1914), and an earlier (1923) paper of Banach as the precursors to their work.
He has also developed numerous pieces of instructional software, including Turing's World, Tarski's World, Fitch, and Hyperproof, software that allows computers to support the reasoning process.
Lemmon was a pioneer of the modern approach to the semantics of modal logic, particularly through his collaboration with Dana Scott, but also became interested in the rival algebraic semantics of modal logic that follows more closely the kind of semantics found in the work of Tarski and Jònsson.
Besides probability, some of these were on curves of minimal length under constraints on curvature and initial and final tangents (see Dubins path), Tarski's circle squaring problem, convex analysis, and geometry.
Semantics of this form has not provided a very great challenge to that sketched in Tarski's Semantic theory of truth, but many philosophers interested in reconstituting the semantics of logic in a way that respects Ludwig Wittgenstein's meaning is use have felt that harmony holds the key.
Montague, one of Tarski's most accomplished American students, spent his entire career teaching in the UCLA Department of Philosophy, where he supervised the dissertations of Nino Cocchiarella and Hans Kamp.
Tarski's lecture at the 1950 International Congress of Mathematicians in Cambridge ushered in a new period in which model-theoretic aspects were developed, mainly by Tarski himself, as well as C.C. Chang, Leon Henkin, Bjarni Jónsson, Roger Lyndon, and others.