# Hilbert's program then and now

@article{Zach2007HilbertsPT, title={Hilbert's program then and now}, author={Richard Zach}, journal={arXiv: Logic}, year={2007}, pages={411-447} }

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Godel’s incompleteness theorems show that the program… Expand

#### 60 Citations

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

- Mathematics
- 2015

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… Expand

Hilberťs Programme and Gödel's Theorems

- Philosophy, Mathematics
- 2005

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.… Expand

Proof Theory

- Mathematics
- 2014

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… Expand

A Gentle Introduction to Membrane Systems and Their Computational Properties

- Computer Science
- Bio-Inspired Computing Models and Algorithms
- 2019

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. Expand

Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic

- Mathematics
- 2015

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… Expand

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

- Mathematics
- Journal of Humanistic Mathematics
- 2019

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… Expand

On A.Ya. Khinchin's paper ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ (1926): A translation with introduction and commentary

- Mathematics
- 2016

Abstract The translation into English of Aleksandr Yakovlevich Khinchin's (1894–1959) 1926 paper entitled ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ is… Expand

Anti-Foundational Categorical Structuralism

- Philosophy
- 2012

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… Expand

Reconstructing Hilbert to Construct Category Theoretic Structuralism

- Mathematics
- 2009

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… Expand

Extensions of the Constructivist Real Number System

- Mathematics
- 2018

The paper reviews the most consequential defects and rectification of traditional mathematics and its foundations. While this work is only the tip of the iceberg, so to speak, it gives us a totally… Expand

#### References

SHOWING 1-10 OF 301 REFERENCES

Hilbert's program

- Mathematics
- 1986

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… Expand

Hilbert's Program Relativized: Proof-Theoretical and Foundational Reductions

- Mathematics, Computer Science
- J. Symb. Log.
- 1988

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. Expand

Hilbert's programs: 1917-1922

- Mathematics, Computer Science
- Bull. Symb. Log.
- 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. Expand

Completeness before Post: Bernays, Hilbert, and the development of propositional logic

- Mathematics, Computer Science
- Bull. Symb. Log.
- 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. Expand

A VARIANT TO HILBERT'S THEORY OF THE FOUNDATIONS OF ARITHMETIC*

- 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)… Expand

Hilbert and the emergence of modern mathematical logic

- Mathematics
- 1997

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… Expand

Hilbert's Program and the Omega-Rule

- Mathematics, Computer Science
- J. Symb. Log.
- 1994

It is shown that Detlefsen's proposal is unacceptable as originally formulated, but that a reasonable modification of the rule he suggests leads to a partial program already studied for many years, and some of the limitations of such programs are determined. Expand

Hilbert's ‘Verunglückter Beweis’, the first epsilon theorem, and consistency proofs

- Mathematics
- 2002

In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's… Expand

Proof-Theoretic Reduction As A Philosopher's Tool

- Philosophy
- 2000

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… Expand

HILBERT'S PROGRAMME

- Philosophy
- 1958

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… Expand