Proof Theory

@inproceedings{Avigad2014ProofT,
  title={Proof Theory},
  author={Jeremy Avigad},
  year={2014}
}
  • J. Avigad
  • Published 6 November 2017
  • Mathematics
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… 

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References

SHOWING 1-10 OF 39 REFERENCES

Hilbert's program then and now

Forcing in Proof Theory

  • J. Avigad
  • Philosophy
    Bulletin of Symbolic Logic
  • 2004
TLDR
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.

Handbook of proof theory

Saturated models of universal theories

Structural proof theory

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

Hilbert's Programs: 1917–1922

  • W. Sieg
  • Philosophy
    Bulletin of Symbolic Logic
  • 1999
TLDR
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.

Fragments of arithmetic

  • W. Sieg
  • Mathematics
    Ann. Pure Appl. Log.
  • 1985

Complexity of Propositional Proofs

TLDR
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.

Number theory and elementary arithmetic

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.