# The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal

@inproceedings{Woodin1999TheAO, title={The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal}, author={W. Hugh Woodin}, year={1999} }

This is the revised edition of a well-established monograph on the identification of a canonical model in which the Continuum Hypothesis is false. Written by an expert in the field, it is directed to researchers and advanced graduate students in Mathematical Logic and Set Theory. The second edition is updated to take into account some of the developments in the decade since the first edition appeared, this includes a revised discussion of ?-logic and related matters.

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## References

SHOWING 1-4 OF 4 REFERENCES

Optimal approximations by piecewise smooth functions and associated variational problems

- Mathematics
- 1989

Abstract : This reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision. In computer vision, a fundamental…

Sulle funzioni a variazione limitata

- Mathematics
- 1936

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