# Forcing in Proof Theory

@article{Avigad2004ForcingIP, title={Forcing in Proof Theory}, author={Jeremy Avigad}, journal={Bulletin of Symbolic Logic}, year={2004}, volume={10}, pages={305 - 333} }

Abstract Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue 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…

## 30 Citations

Provably recursive functions of constructive and relatively constructive theories

- MathematicsArch. Math. Log.
- 2010

Conservation theorems for theories of classical first-order arithmetic over their intuitionistic version are proved and generalized conservation results for intuitionistic theories when certain weak forms of the principle of excluded middle are added to them are proved.

Mechanization of Separation in Generic Extensions

- EconomicsArXiv
- 2019

Paulson's library on constructibility is extended with renaming of variables for internalized formulas, improved results on definitions by recursion on well-founded relations, and sharpened hypotheses in his development of relativization and absoluteness.

Finitely Axiomatized Theories Lack Self-Comprehension

- Mathematics
- 2021

In this paper we prove that no consistent finitely axiomatized theory one-dimensionally interprets its own extension with predicative comprehension. This constitutes a result with the flavor of the…

A new model construction by making a detour via intuitionistic theories I: Operational set theory without choice is Π1-equivalent to KP

- PhilosophyAnn. Pure Appl. Log.
- 2015

A nominal exploration of intuitionism

- MathematicsCPP
- 2016

This papers extends the Nuprl proof assistant with named exceptions and handlers, as well as a nominal fresh operator, to prove a version of Brouwer's Continuity Principle for numbers and provides a simpler proof of a weaker version of this principle that only uses diverging terms.

Forcing and Type Theory

- MathematicsCSL
- 2009

The technique of forcing is presented from a constructive point of view and the connections with proof theory and constructive mathematics are not so surprising.

A Kuroda-style j-translation

- PhilosophyArch. Math. Log.
- 2019

It is shown that there exists a similar translation of intuitionistic logic into itself which is more in the spirit of Kuroda's negative translation, where the key is to apply the nucleus not only to the entire formula and universally quantified subformulas, but to conclusions of implications as well.

Constructive forcing, CPS translations and witness extraction in Interactive realizability †

- Computer ScienceMathematical Structures in Computer Science
- 2015

A constructive and efficient method is introduced based on a new ‘(state-extending-continuation)-passing-style translation’ whose properties are described with the constructive forcing/reducibility semantics.

Forcing as a Program Transformation

- Computer Science2011 IEEE 26th Annual Symposium on Logic in Computer Science
- 2011

It is shown how to avoid the cost of the transformation by introducing an extension of Krivine's abstract machine devoted to the execution of proofs constructed by forcing, which induces new classical realizability models and present the corresponding adequacy results.

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