150 Logic as a Tool some of the complications entailed by Bertrand Russell’s theory of types, and would at the same time avoid the well-known paradoxes, in particular the Russell paradox.” While it later turned out that the original system of lambda-calculus did contain an inconsistency, the idea flourished and became very influential in computer science as the basis of functional programming In 1936 Church published two papers of fundamental importance to logic and computability theory In the first, An Unsolvable Problem of Elementary Number Theory, he defined the notion of “effective calculability” by identifying it with the notion of recursive function, or – as it had turned out to be equivalent – of the lambda-definable function of natural numbers He then proved that the problem of deciding whether a given first-order formula of the language with addition and multiplication is true of the arithmetic of natural numbers is not effectively calculable By equating effective calculability with algorithmic solvability, Church put forward what was later called by Kleene Church’s Thesis Church essentially proved that the truth in the elementary arithmetic is algorithmically unsolvable In his other historic 1936 paper, A Note on the Entschiedungsproblem (just two pages long, followed by a two-page correction), Church proved what is now called Church’s Undecidability Theorem: the problem of deciding validity of formulas in first-order logic is not effectively calculable, which means – assuming Church’s Thesis – it is algorithmically unsolvable Church had an extremely long and fruitful scientific career, spanning over more than 70 years since his first publication as an undergraduate student in 1924 until his last paper, A Theory of the Meaning of Names, published in 1995 In 1956 he published his classic textbook Introduction to Mathematical Logic, which educated and influenced generations of logicians Church was also an extremely popular and successful supervisor and had 31 doctoral students, many of whom became distinguished logicians and computer scientists, including Martin Davis, Leon Henkin, John Kemeny, Steven Kleene, Michael O Rabin, Nicholas Rescher, Hartley Rogers, J Barkley Rosser, Dana Scott, Reymond Smullyan, and Alan Turing Stephen Cole Kleene (05.01.1909–25.01.1994) was an American mathematician and logician, the founder (along with Church, Gödel, Turing, and Post) of recursion theory, one of the main branches of mathematical logic, providing the logical foundations of the theory of algorithms and computability and of theoretical computer science in general Kleene received a PhD in mathematics from Princeton University in 1934, for the thesis A Theory of Positive Integers in Formal Logic supervised by Alonzo Church He also made important contributions to Church’s lambda-calculus In 1935, Kleene joined the Mathematics Department at the University of Wisconsin-Madison, where he spent most of his academic career In 1939–40, while he was a visiting scholar at the Institute for