Papers by Solomon Feferman
... PROOF THEORETIC EQUIVALENCES BETWEEN CLASSICAL AND CONSTRUCTIVE THEORIES FOR ANALYSIS by Solo... more ... PROOF THEORETIC EQUIVALENCES BETWEEN CLASSICAL AND CONSTRUCTIVE THEORIES FOR ANALYSIS by Solomon Feferman and Wilfried Sieg Page 2. 79 INTRODUCTION ==~== Hilbert's program of justifying all of mathematics in finitist terms cannot ...
A notion of finitary inductively presented (f.i.p.) logic is proposed here, which includes all sy... more A notion of finitary inductively presented (f.i.p.) logic is proposed here, which includes all syntactically described logics (formal systems) met in practice. A f.i.p. theory FS 0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS 0 is a conservative extension of PRA. The aims of this work are (i) conceptual, (ii) pedagogical and (iii) practical. The system FS 0 serves under (i) and (ii) as a theoretical framework for the formalization of metamathematics. The general approach may be used under (iii) for the computer implementation of logics. In all cases, the work aims to make the details manageable in a natural and direct way.
Predicative mathematics in the sense originating with Poincaré and Weyl begins by taking the natu... more Predicative mathematics in the sense originating with Poincaré and Weyl begins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or set-theoretic standpoint, this appears problematic, for, as the story is usually told, impredicative principles seem to play an essential role in the foundations of arithmetic itself. 1 It is the main purpose of this paper to show that this appearance is illusory: as will emerge, a predicatively acceptable axiomatization of the natural number system can be formulated, and both the existence of structures of the relevant type and the categoricity of the relevant axioms can be proved in a predicatively acceptable way.
In his book Shadows of the Mind: A search for the missing science of consciousness [SM below], Ro... more In his book Shadows of the Mind: A search for the missing science of consciousness [SM below], Roger Penrose has turned in another bravura performance, the kind we have come to expect ever since The Emperor's New Mind [ENM ] appeared. In the service of advancing his deep convictions and daring conjectures about the nature of human thought and consciousness, Penrose has once more drawn a wide swath through such topics as logic, computation, artificial intelligence, quantum physics and the neuro-physiology of the brain, and has produced along the way many gems of exposition of difficult mathematical and scientific ideas, without condescension, yet which should be broadly appealing. 1 While the aims and a number of the topics in SM are the same as in ENM , the focus now is much more on the two axes that Penrose grinds in earnest. Namely, in the first part of SM he argues anew and at great length against computational models of the mind and more specifically against any account of mathematical thought in computational terms. Then in the second part, he argues that there must be a scientific account of consciousness but that will require a (still to be found) non-computational extension or modification of present-day quantum physics.
Questions of definedness are ubiquitous in mathematics. Informally, these involve reasoning about... more Questions of definedness are ubiquitous in mathematics. Informally, these involve reasoning about expressions which may or may not have a value. This paper surveys work on logics in which such reasoning can be carried out directly, especially in computational contexts. It begins with a general logic of "partial terms", continues with partial combinatory and lambda calculi, and concludes with an expressively rich theory of partial functions and polymorphic types, where termination of functional programs can be established in a natural way.
In this paper we specialize the notion of abstract computational procedure previously introduced ... more In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functionals of type level ≤ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type.
Hilbert's program modi ed.
1 Cf. in G odel 1986] the items dated: 420-425). 3 The kind of proposition in question is sometim... more 1 Cf. in G odel 1986] the items dated: 420-425). 3 The kind of proposition in question is sometimes referred to by G odel as being of \Goldbach t ype" i.e. in 0 1 form, and sometimes as one concerning solutions of Diophantine equations, of the form (P)D = 0 , w h e r e P is a quanti er expression with variables ranging over the natural numbers cf. more speci cally, the lecture notes *193? in G odel 1995].
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Papers by Solomon Feferman