Infinity and Truth

The following sections are included:

• Contents
• Foreword
• Preface

The infinite in mathematics has two manifestations. Its occurrence in analysis has been satisfactorily formalized and demystified by the ε-δ method of Bolzano, Cauchy and Weierstrass. It is of course the ‘set-theoretic infinite’ that concerns me here. Once the existence of an infinite set is accepted, the axioms of set theory imply the existence of a transfinite hierarchy of larger and larger orders of infinity. I shall review some well-known facts about the influence of these axioms of infinity to the everyday mathematical practice and point out to some, as of yet not understood, phenomena at the level of the third-order arithmetic. Technical details from both set theory and operator algebras are kept at the bare minimum. In the Appendix, I include definitions of arithmetical and analytical hierarchies in order to make this paper more accessible to non-logicians. In this paper I am taking a position intermediate between pluralism and non-pluralism (as defined by P. Koellner in the entry on large cardinals and determinacy of the Stanford Encyclopaedia of Philosophy) with an eye for applications outside of set theory.

I shall argue that the commonly held V ≠ L via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-theoretic universes.

Those who wish to teach in the lycées and collèges of France must obtain a Certificat d'Aptitude au Professorat de l'Enseignement du Second Degré, commonly called the CAPES. For that certificate, the first hurdle to be passed is a written examination. In at least ten recent years the syllabus for the mathematics section of that examination, as specified in the relevant special numbers of the Bulletin Officiel, has contained the line…

We present some ideas in furtherance of objectivity in mathematics. We call for closer integration of mathematics with the rest of human knowledge. We note some insights which can be drawn from current research programs in the foundations of mathematics.

I have argued elsewhere [8] that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by reliance on ad hoc large domain assumptions. In this chapter I argue that sort logic, a powerful extension of second order logic, provides a foundation for mathematics without any ad hoc large domain assumptions. The large domain assumptions are replaced by ZFC-like axioms. Despite this resemblance to set theory sort logic retains the structuralist approach to mathematics characteristic of second order logic. As a model-theoretic logic sort logic is the strongest logic. In fact, every model class definable in set theory is the class of models of a sentence of sort logic. Because of its strength sort logic can be used to formulate particularly strong reflection principles in set theory.

We find that second order quantification is problematic when a quantified concept variable is supposed to function predicatively. This issue is analyzed and it is shown that a constructive interpretation of the falling under relation suffices to resolve the difficulty. We are then able to present a formal system for reasoning about concepts. We prove that this system is consistent and we investigate the extent to which it is able to interpret set theoretic and number theoretic systems of a more standard type.

In this chapter we present an analysis of uses of infinity in “applied mathematics”, by which we mean mathematics as a tool for understanding the real world (whatever the latter means). This analysis is based on certain developments in Model Theory, and lessons and question related to these developments.

This document is a record of my contribution to a panel discussion which took place on July 27, 2011 as part of the Infinity and Truth Workshop held at the Institute for Mathematical Sciences, National University of Singapore, July 25–29, 2011.

The following sections are included:

• Introduction
• Questions
  • Ilijas Farah
  • Moti Gitik
  • Joel David Hamkins
  • Juliette Kennedy
  • Steffen Lempp
  • Stephen G. Simpson
  • Theodore Slaman
  • Jouko Väänänen
  • Nik Weaver
  • W. Hugh Woodin
  • Boris Zilber
• References