Predicativity

1 What is predicativity? While the term suggests that there is a single idea involved, what the history will show is that there are a number of ideas of predicativity which may lead to different logical analyses, and I shall uncover these only gradually. A central question will then be what, if anything, unifies them. Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and so I shall give them special attention. NB. Ahistorically, modern logical and set-theoretical notation will be used throughout, as long as it does not conflict with original intentions. To begin with, the terms predicative and non-predicative (later, impredica-tive) were introduced by Russell (1906) in his struggles dating from 1901 to carry out the logicist program in the face of the set-theoretical paradoxes. Rus-sell called a propositional function ϕ(x) predicative if it defines a class, i.e., if the class {x : ϕ(x)} exists, and non-predicative otherwise. Thus, for example, the propositional function x ∈ x figuring in Russell's paradox is impredicative. Since the admission of classes defined by arbitrary propositional functions in Frege's execution of his logicist program led to its demise as a result of this paradox, if the program were to be resurrected, it would somehow have to incorporate a criterion for distinguishing predicative from impredicative ones. Russell's first attempts to separate these were highly uncertain, and it was only through the engagement of Henri Poincaré in the problem starting in his article (1906) that progress began to be made. Poincaré took several paradoxes as examples to try to elicit what was common to them, namely the Burali-Forti paradox of the largest ordinal number, König's paradox of the least non-definable ordinal 1 The subject of predicativity is one that has been of great interest to me and has periodically commanded much of my attention over the last forty years. It involves substantial developments in logic and mathematics and is of significance for the philosophy of mathematics. However, it is still unsettled how best to assess these various aspects of predicativity. On March 28, 2002, for a joint meeting of the American Philosophical Association and the Association for Symbolic Logic, Jeremy Avigad, Geoffrey Hellman and I participated in a symposium organized by Paolo Mancosu entitled " Predicativity: Problems and Prospects. " In my lecture I concentrated on the …

[1]  Michael Louis Corrado,et al.  Ontology and the Vicious Circle Principle , 1974 .

[2]  Solomon Feferman,et al.  Persistent and invariant formulas for outer extensions , 1968 .

[3]  Solomon Feferman,et al.  The unfolding of non-finitist arithmetic , 2000, Ann. Pure Appl. Log..

[4]  Solomon Feferman,et al.  Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics , 1992, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.

[5]  Hao Wang,et al.  The Formalization of Mathematics , 1954, J. Symb. Log..

[6]  Solomon Feferman,et al.  Gödel's program for new axioms: why, where, how and what? , 1996 .

[7]  P. Lorenzen.  Einführung in die operative Logik und Mathematik , 1955 .

[8]  E. Zermelo.  Neuer Beweis für die Möglichkeit einer Wohlordnung , 1907 .

[9]  Stephen G. Simpson,et al.  Set Theoretic Aspects of Atr0 , 1982 .

[10]  S. Feferman,et al.  Challenges to Predicative Foundations of Arithmetic , 2000 .

[11]  B. Russell.  Mathematical Logic as Based on the Theory of Types , 1908 .

[12]  Geoffrey Hellman,et al.  Predicative foundations of arithmetic , 1995, J. Philos. Log..

[13]  Solomon Feferman,et al.  Reflecting on incompleteness , 1991, Journal of Symbolic Logic.

[14]  K. Gödel.  Consistency-Proof for the Generalized Continuum-Hypothesis. , 1939, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Gérard G. Emch,et al.  Algebraic methods in statistical mechanics and quantum field theory , 1972 .

[16]  D. Scott.  Review: G. Kreisel, Ordinal Logics and the Characterization of Informal Concepts of Proof , 1962 .

[17]  Jean-Baptiste Lully,et al.  The collected works , 1996 .

[18]  Clifford Spector,et al.  Recursive well-orderings , 1955, Journal of Symbolic Logic.

[19]  Solomon Feferman,et al.  Impredicativity of the existence of the largest divisible subgroup of an Abelian p-group , 1975 .

[20]  Andrew P. Arana.  In the light of logic , 1998 .

[21]  Gerhard Jäger,et al.  Systems of Explicit Mathematics with Non-Constructive µ-Operator, Part II , 1996, Ann. Pure Appl. Log..

[22]  G. B. M..  Principia Mathematica , 1911, Nature.

[23]  Feng Ye.  Strict constructivism and the philosophy of mathematics , 2000 .

[24]  J. Heijenoort.  From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 , 1967 .

[25]  G. Kreisel,et al.  Principles of Proof and Ordinals Implicit in Given Concepts , 1970 .

[26]  Paul Lorenzen,et al.  Constructive definition of certain analytic sets of numbers , 1959, Journal of Symbolic Logic.

[27]  J. Paris.  A Mathematical Incompleteness in Peano Arithmetic , 1977 .

[28]  Solomon Feferman,et al.  Systems of predicative analysis , 1964, Journal of Symbolic Logic.

[29]  Penelope Maddy,et al.  Indispensability and Practice , 1992 .

[30]  Gerhard Jäger,et al.  Systems of Explicit Mathematics with Non-Constructive µ-Operator, Part I , 1993, Ann. Pure Appl. Logic.

[31]  W. Goldfarb.  Poincaré against the logicists , 1988 .

[32]  Paolo Mancosu,et al.  From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s , 1997 .