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The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions coexist. Clearly neither the algebra of logic nor the logicist’s logic is identical to the present form of mathematical logic, yet some of their basic ideas can be distinctly recognized within it. One of such ideas is Boole’s view that logic is the study of the laws of thought. This is not to be meant in a psychologistic way. Frege himself states that the task of logic can be represented “as the investigation of the mind; [though] of the mind, not of minds” [17, p. 369]. Moreover Frege never charges Boole with being psychologistic and in a letter to Peano even distinguishes between the followers of Boole and “the psychological logicians ” [16, p. 108]. In fact for Boole the laws of thought which are the object of logic belong “to the domain of what is termed necessary truth ” [2, p. 404]. For him logic does not depend on psychology, on the contrary psychology depends on logic insofar as it is only through an investigation of logical operations that we could obtain “some probable

© This paper is not for reproduction without the express permission of the author. Information is understood as a mediated construction, the Sign, which is a cohesion of meaning, both material and conceptual, derived from the measurement within spatial and temporal parameters of energy/matter by six predicate relations. Three predicate relations enforce states; three predicate relations enforce dynamics. The Sign is understood as a triadic cohesion of three predicates. 1

The objective of this paper is to introduce an ordinal representation system which has been employed in the determination of the proof--theoretic strength of \Pi 1 2 comprehension and related systems. 1 Introduction The purpose of this paper is to provide an ordinal representation system for the system of \Pi 1 2 analysis, which is the subsystem of formal second order arithmetic, Z 2 , with comprehension confined to \Pi 1 2 -formulae. The ordinal representation can also be used to provide ordinal analyses for theories that are reducible to iterated \Pi 1 2 comprehension, e.g. \Delta 1 3 comprehension. The details will be laid out in the second part of this paper. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z 2 . Considerable progress has been made...

The objective of this paper is to present an ordinal analysis for the fragment of second order arithmetic with \Delta 1 2 -comprehension, bar induction and \Pi 1 2 -comprehension for formulae without set parameters.

There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper