Kurt Gödel and Computability Theory

Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s

Max Dehn, Axel Thue, and the Undecidable

This is a short essay on the roles of Max Dehn and Axel Thue in the formulation of the word problem for (semi)groups, and the story of the proofs showing that the word problem is undecidable.Comment: Definition of undecidability and unsolvability improve

The paradox of idealization

A well-known proof by Alonzo Church, first published in 1963 by Frederic Fitch, purports to show that all truths are knowable only if all truths are known. This is the Paradox of Knowability. If we take it, quite plausibly, that we are not omniscient, the proof appears to undermine metaphysical doctrines committed to the knowability of truth, such as semantic antirealism. Since its rediscovery by Hart and McGinn (1976), many solutions to the paradox have been offered. In this article, we present a new proof to the effect that not all truths are knowable, which rests on different assumptions from those of the original argument published by Fitch. We highlight the general form of the knowability paradoxes, and argue that anti-realists who favour either an hierarchical or an intuitionistic approach to the Paradox of Knowability are confronted with a dilemma: they must either give up anti-realism or opt for a highly controversial interpretation of the principle that every truth is knowable

LEVEL THEORY, PART 3: A BOOLEAN ALGEBRA OF SETS ARRANGED IN WELL-ORDERED LEVELS

On a very natural image of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surr

A Conversation with Leo Goodman

Leo A. Goodman was born on August 7, 1928 in New York City. He received his A.B. degree, summa cum laude, in 1948 from Syracuse University, majoring in mathematics and sociology. He went on to pursue graduate studies in mathematics, with an emphasis on mathematical statistics, in the Mathematics Department at Princeton University, and in 1950 he was awarded the M.A. and Ph.D. degrees. His statistics professors at Princeton were the late Sam Wilks and John Tukey. Goodman then began his academic career as a statistician, and also as a statistician bridging sociology and statistics, with an appointment in 1950 as assistant professor in the Statistics Department and the Sociology Department at the University of Chicago, where he remained, except for various leaves, until 1987. He was promoted to associate professor in 1953, and to professor in 1955. Goodman was at Cambridge University in 1953--1954 and 1959--1960 as visiting professor at Clare College and in the Statistical Laboratory. And he spent 1960--1961 as a visiting professor of mathematical statistics and sociology at Columbia University. He was also a research associate in the University of Chicago Population Research Center from 1967 to 1987. In 1970 he was appointed the Charles L. Hutchinson Distinguished Service Professor at the University of Chicago, a title that he held until 1987. He spent 1984--1985 at the Center for Advanced Study in the Behavioral Sciences in Stanford. In 1987 he was appointed the Class of 1938 Professor at the University of California, Berkeley, in the Sociology Department and the Statistics Department. Goodman's numerous honors include honorary D.Sc. degrees from the University of Michigan and Syracuse University, and membership in the National Academy of Sciences, the American Academy of Arts and Sciences, and the American Philosophical Society.Comment: Published in at http://dx.doi.org/10.1214/08-STS276 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church's intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms of Church's intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin's intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi

Population, Poverty, Politics and the Reproductive Health Bill

Following an earlier paper titled “Population and Poverty: The Real Score” (UPSE Discussion Paper 0415, December 2004), the present paper was first issued in August 2008 as a contribution to the public debate on the population issue that never seemed to die in this country. The debate heated up about that time in reaction to a revival of moves to push for legislation on reproductive health and family planning (RH/FP). Those attempts at legislation, however, failed in the 13th Congress, and again in the 14th Congress. Since late last year, the debate has been heating up further on the heels of President Noy Aquino’s pronouncements seeming to favor RH/FP, though he prefers the nomenclature “responsible parenthood”. With some updating of the data, this paper remains as relevant as ever to the ongoing public debate. It is being re-issued as a Discussion Paper for wider circulation.population, reproductive health, poverty, Philippines