# Hilbert’s Finitism: Historical, Philosophical, and Metamathematical Perspectives

Dissertation, University of California, Berkeley, Spring 2001

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing—using only so-called finitistic principles—that these formalizations are free of contradictions.

In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917-18, and that Bernays’s contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert’s Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann’s 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to $$\omega^{\omega^\omega}$$. He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert’s epsilon-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert’s school.

In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert’s finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann’s dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert’s conception of finitism is concerned. Tait’s conception, however, differs from Hilbert’s in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning.

## Note

Chapter 2 appeared as Completeness before Post; the version here corrects some misprints and contains a few minor additions. A revised version of Chapter 3 is The practice of finitism.

 1 Introduction 1 1.1 David Hilbert and the Foundations of Mathematics 1 1.2 Hilbert, Bernays, and Logic 4 1.3 Hilbert’s Proof-Theoretical Program 7 1.4 Finitism 11 2 Completeness before Post: Bernays, Hilbert, and the Development of Propositional Logic 15 2.1 Introduction 15 2.2 Semantics, Normal Forms, Completeness 17 2.2.1 Prehistory: Hilbert’s Lectures on Logical Principles of Mathematical Thought 1905 17 2.2.2 The Structure of Prinzipien der Mathematik 21 2.2.3 The Propositional Calculus 23 2.2.4 Consistency and Completeness 25 2.2.5 The Contribution of Bernays’s Habilitationsschrift 27 2.2.6 A Brief Comparison with Post’s thesis 29 2.3 Hilbert or Bernays? 31 2.4 Dependence and Independence 35 2.5 Axioms and Rules 36 2.6 Lasting Influences 41 3 The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert’s Program 56 3.1 Introduction 56 3.2 Early Consistency Proofs 58 3.2.1 The Propositional Calculus and the Calculus of Elementary Computation 61 3.2.2 Elementary Number Theory with Recursion and Induction Rule 66 3.2.3 The $$\varepsilon$$-Calculus and the Axiomatization of Mathematics 68 3.3 Ackermann’s Dissertation 73 3.3.1 Second-order Primitive Recursive Arithmetic 75 3.3.2 The Consistency Proof for the System without Epsilons 77 3.3.3 Ordinals, Transfinite Induction, and Finitism 83 3.3.4 The $$\varepsilon$$-Substitution Method 85 3.3.5 Assessment and Complications 91 3.4 Conclusion 94 4 Finitism and Mathematical Knowledge 109 4.1 Introduction 109 4.2 The Significance of Finitism 111 4.3 Finitism in Hilbert 114 4.3.1 Numbers and Numerals 114 4.3.2 Statements 121 4.3.3 Finitistic Functions 125 4.3.4 Finitism and PRA 128 4.4 The Status of Finitism 132 4.5 Parsons’ Criticism of Finitism as Intuitive Knowledge 134 4.6 Tait’s Analysis of Finitism 141 4.7 Conclusion 147 Bibliography 155