Tarski's theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic

Although the theory of definability had many important antecedents-such as the descriptive set theory initiated by the French semi-intuitionists in the early 1900s-the main ideas were first laid out in precise mathematical terms by Alfred Tarski beginning in 1929. We review here the basic notions of languages, explicit definability, and grammatical complexity, and emphasize common themes in the theories of definability for four important languages underlying, respectively, descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic. We review the history of previous studies of the similarities and differences in the theories of definability of the first three of these four languages. A seminal role leading toward unification of the theories has been played by the separation principles introduced (and regarded as so 4 ''fundamental") by Nikolai Luzin in 1927. Emphasizing analogies and driving toward further unification embracing finite-universe logic (a setting of special interest in theoretical computer science) we concentrate on a simple example-the first and second separation principles for existential-universal first-order sentences (which are known to fail in descriptive set theory, recursive function theory, and classical pure logic). Using this as a test case for the fundamental problem of how to "finitize" arguments in classical pure logic to the finite-universe case, we are led to the analogous negative solution by using the theory of certain special graphs: a graph is (m, n, p, q)-special for any positive integers m,n, p, q iff it is bipartite with m red points and n blue points and for every p-tuple (q-tuple) of red (blue) points there is a blue (red) point to (with) which they are all connected (disconnected). As an aside we introduce for further study a natural "Ramseyesque" increasing sequence A of positive integers, where A(p) is the least positive integer n for which an (n, n, p, p)-special graph exists. (C) 2003 Elsevier B.V. All rights reserved.