# Proof Theory

@inproceedings{Avigad2014ProofT, title={Proof Theory}, author={Jeremy Avigad}, year={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 those systems consistent using restricted, finitary means. The program thus viewed mathematics as a system of reasoning with precise linguistic norms, governed by rules that can be described and studied in concrete terms. Today such a viewpoint has applications in mathematics, computer science, and the…

## 237 Citations

### On the Proof Theory of Infinitary Modal Logic

- Mathematics, PhilosophyStudia Logica
- 2022

The article deals with infinitary modal logic. We first discuss the difficulties related to the development of a satisfactory proof theory and then we show how to overcome these problems by…

### XV—On Consistency and Existence in Mathematics

- Philosophy
- 2020

This paper engages the question Does the consistency of a set of axioms entail the existence of a model in which they are satisfied? within the frame of the Frege-Hilbert controversy. The question is…

### Proof theory and proof systems for projective and affine geometry

- Mathematics

Purpose of this research area is the development of formal methods and tools to deal with various geometries. Furthermore we aim at a formal description of sketches in these geometries, which seem to…

### University of Birmingham A cut-free cyclic proof system for kleene algebra

- Mathematics, Computer Science
- 2017

A sound non-wellfounded proof system whose regular (or ‘cyclic’) proofs are complete for (in)equations between regular expressions, achieved by using hypersequents rather than usual sequents, and relying on the discreteness of rational languages to drive proof search.

### SOME THEORIES WITH POSITIVE INDUCTION OF ORDINAL STRENGTH <pcoO

- Mathematics

. This paper deals with: (i) the theory ID* which results from ID] by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON(^)…

### Consistency, Truth and Existence

- Philosophy
- 2013

In the first section we mention the main philosophical approaches in the standard classification: platonism, intuitionism, logicism and formalism. In the second section, we discuss consistency…

### A Cut-Free Cyclic Proof System for Kleene Algebra

- Mathematics, Computer ScienceTABLEAUX
- 2017

A sound non-wellfounded proof system whose regular (or ‘cyclic’) proofs are complete for (in)equations between regular expressions, achieved by using hypersequents rather than usual sequents, and relying on the discreteness of rational languages to drive proof search.

### A Proof Theory for the Logic of Provability in True Arithmetic

- MathematicsStud Logica
- 2020

A proof theory for GLS is developed based on the sequent calculus method and the cut-elimination is proved and the equivalence of GL and GLS with respect to a special form of formulas which are called Gödel sentences is proved using a purely proof-theoretical method.

### Probabilistic Extensions of Various Logical Systems

- Philosophy
- 2020

This chapter presents the logic LFOP1 which may be suitable to formalize reasoning about degrees of beliefs. The aim is that this chapter serves as an illustration for syntax, semantics and the main…

### Intermediate Goodstein principles

- Mathematics
- 2020

The original Goodstein process proceeds by writing natural numbers in nested exponential k -normal form, then successively raising the base to k + 1 and subtracting one from the end result. Such…

## References

SHOWING 1-10 OF 39 REFERENCES

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- PhilosophyBulletin of Symbolic Logic
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It is argued that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms.

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- Mathematics
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From natural deduction to sequent calculus, diversity and unity in structural proof theory are explored and the quantifiers are explained.

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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.

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Propositional proof complexity is extremely well connected to very different disciplines like computational complexity, theoretical cryptography, automated theorem proving, mathematical logic, algebra and geometry, and methods and concepts employed in the area are also very diverse.

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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function.…