this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.

A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie " - the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency - technical results in pro...

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...

this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of #

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.

We must remember that in Hilbert’s time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects... The contrast with today’s foggy atmosphere of intellectual exhaustion and compartmentalization could not be more striking. — Stephen Simpson ([15], p. 350) From the standpoint of mainstream mathematics, the great foundational debates of the early twentieth century were decisively settled in favor of Cantorian set theory, as formalized in the system ZFC (Zermelo-Fraenkel set theory including the axiom of choice). Although basic foundational questions have never entirely disappeared, it seems fair to say that they have retreated to the periphery of mathematical practice. Sporadic alternative proposals like topos theory or Errett Bishop’s constructivism have never attracted a substantial mainstream following, and Cantor’s universe is generally acknowledged as the arena in which modern mathematics takes place. Despite this history, I believe a strong case can be made for abandoning Cantorian

We must remember that in Hilbert’s time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects... The contrast with today’s foggy atmosphere of intellectual exhaustion and compartmentalization could not be more striking. — Stephen Simpson ([14], p. 350) From the standpoint of mainstream mathematics, the great foundational debates of the early twentieth century were decisively settled in favor of Cantorian set theory, as formalized in the system ZFC (Zermelo-Fraenkel set theory including the axiom of choice). Although basic foundational questions have never entirely disappeared, it seems fair to say that they have retreated to the periphery of mathematical practice. Sporadic alternative proposals like topos theory or Errett Bishop’s constructivism have never attracted a substantial mainstream following, and Cantor’s universe is generally acknowledged as the arena in which modern mathematics takes place. Despite this history, I believe a strong case can be made for abandoning Cantorian