Hilbert's program then and now

  title={Hilbert's program then and now},
  author={Richard Zach},
  journal={arXiv: Logic},
  • R. Zach
  • Published 29 August 2005
  • Philosophy
  • arXiv: Logic

The Use of Trustworthy Principles in a Revised Hilbert’s Program

After the failure of Hilbert’s original program due to Godel’s second incompleteness theorem, relativized Hilbert’s programs have been suggested. While most metamathematical investigations are

Hilberťs Programme and Gödel's Theorems

In this paper, we attempt to show that a weak version of Hilberťs metamathematics is compatible with Godel's Incompleteness Theorems by employing only what are clearly natural provability predicates.

Proof Theory

Proof theory began in the 1920’s as a part of Hilbert’s program, which aimed to secure the foundations of mathematics by modeling infinitary mathematics with formal axiomatic systems and proving

A Gentle Introduction to Membrane Systems and Their Computational Properties

The theory of computation investigates the nature and properties of algorithmic procedures and the development of computational complexity theory, pioneered by Hartmanis and Stearns in the paper On the computational complexity of algorithms, that also gives the name to the field.

Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic

Hilbert’s choice operators τ and e, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker

From Solvability to Formal Decidability: Revisiting Hilbert’s “Non-Ignorabimus”

The topic of this article is Hilbert’s axiom of solvability, that is, his conviction of the solvability of every mathematical problem by means of a finite number of operations. The question of

Anti-Foundational Categorical Structuralism

The aim of this dissertation is to outline and defend the view here dubbed “anti-foundational categorical structuralism” (henceforth AFCS). The program put forth is intended to provide an answer the

Reconstructing Hilbert to Construct Category Theoretic Structuralism

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to

Husserl, Cantor & Hilbert: La Grande Crise des Fondements Mathematiques du XIXeme Siecle

Three thinkers of the 19th century revolutionized the science of logic, mathematics, and philosophy. Edmund Husserl (1859-1938), mathematician and a disciple of Karl Weierstrass, made an immense

The Identity Problem: The Case of The Non-trivial Automorphism

This project describes a solution to a problem in Stewart Shapiro’s ante rem structural­ ism, a theory in the philosophy of mathematics. Shapiro’s theory proposes that the nature of mathematical



Hilbert's program

In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls

Hilbert's program relativized; Proof-theoretical and foundational reductions

Here a body of proof-theoretical results stemming from H.P. are surveyed in a way that is closely tied to various reductive foundational aims, albeit going beyond those advanced by Hilbert.

Hilbert's Programs: 1917–1922

  • W. Sieg
  • Philosophy
    Bulletin of Symbolic Logic
  • 1999
The connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century is sketched, the work that laid the basis of modern mathematical logic is described, and the first steps in the new subject of proof theory are analyzed.

Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic

  • R. Zach
  • Philosophy
    Bulletin of Symbolic Logic
  • 1999
It is argued that truth-value semantics, syntactic (“Post-”) and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged.


  • G. Kreisel
  • Mathematics
    The British Journal for the Philosophy of Science
  • 1953
IN Hilbert's theory of the foundations of any given branch of mathematics the main problem is to establish the consistency (of a suitable formalisation) of this branch. Since the (intuitionist)

Hilbert and the emergence of modern mathematical logic

Hilbert's unpublished 1917 lectures on logic, analyzed here, are the beginning of modern metalogic. In them he proved the consistency and Post-completeness (maximal consistency) of propositional

Beyond Hilbert’s Reach?

Work in the foundations of mathematics should provide systematic frameworks for important parts of the practice of mathematics, and the frameworks should be grounded in conceptual analyses that

Proof-Theoretic Reduction As A Philosopher's Tool

1. PROOF-THEORETIC REDUCTION AND HILBERT ’ S PROGRAM Hilbert’s program in the philosophy of mathematics comes in two parts. One part is a technical part. To carry out this part of the program one has


Hilbert's plan for understanding the concept of infinity required the elimination of non-finitist machinery from proofs of finitist assertions. The failure of the original plan leads to a hierarchy

Reflections on Hilbert’s Program

Hilbert’s Program deals with the foundations of mathematics from a very special perspective; a perspective that stems from Hubert’s answer to the question “What Is Mathematics?”. The popular version