# 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…

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## 30 Citations

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## References

SHOWING 1-10 OF 187 REFERENCES

Two applications of Boolean models

- Mathematics, Computer ScienceArch. Math. Log.
- 1998

It is shown here how to give similar arguments, valid constructively, by using Boolean models, a slight variation of ordinary first-order models, where truth values are now regular ideals of a given Boolean algebra.

Continuous Truth I Non-constructive Objects

- Computer Science
- 1984

A general theory of the logic of potentially infinite objects, derived from a theory of meaning for statements concerning these objects, is given, showing that general principles of continuity, local choice and local compactness hold for these models.

Handbook of proof theory

- Mathematics
- 1998

An introduction to proof theory, S.R. Buss first-order proof theory of arithmetic, S.R. Buss hierarchies of provably recursive functions, M. Fairtlough and S.S. Wainer subsystems of set theory and…

Transfer principles in nonstandard intuitionistic arithmetic

- Mathematics, Computer ScienceArch. Math. Log.
- 2002

A term-model construction is presented that assigns a model to any first-order intuitionistic theory, enabling it to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic and showing that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative over primitive recursive arithmetic.

Algebraic proofs of cut elimination

- Mathematics, Computer ScienceJ. Log. Algebraic Methods Program.
- 2001

Algebraic proofs of the cut-elimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a…

Constructive Topology and Combinatorics

- Mathematics, Computer ScienceConstructivity in Computer Science
- 1991

The proofs obtained by this method are well-suited for mechanisation in interactive proof systems that allow the user to introduce inductively defined notions, such as NuPrl, or Martin-Lof set theory.

Formalizing Forcing Arguments in Subsystems of Second-Order Arithmetic

- Mathematics, Computer ScienceAnn. Pure Appl. Log.
- 1996

This method is used to sharpen the conservation theorems of Harrington and Brown-Simpson, giving an effective proof that WKL+0 is conservative over RCA0 with no significant increase in the lengths of proofs.

A uniform semantic proof for cut-elimination and completeness of various first and higher order logics

- Computer Science, MathematicsTheor. Comput. Sci.
- 2002

A natural generalization of Girard's (first order) phase semantics of linear logic to intuitionistic and higher-order phase semantics is presented, which allows for a uniform semantic proof of the cut-elimination theorem and a higher order phase-semantic completeness theorem for various different logical systems at the same time.

Bounded arithmetic, propositional logic, and complexity theory

- Mathematics, Computer ScienceEncyclopedia of mathematics and its applications
- 1995

This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author…

Saturated models of universal theories

- Mathematics, Computer ScienceAnn. Pure Appl. Log.
- 2002

Abstract A notion called Herbrand saturation is shown to provide the model-theoretic analogue of a proof-theoretic method, Herbrand analysis, yielding uniform model-theoretic proofs of a number of…