A formula is true in a model if it has a realizer in the model, and a formula is valid if it has a realizer in every model.

References

Steven Cole Kleene, on the interpretation of intuitionistic number theory, JSL, 10:109-124, 1945

Igor Privara, Branislav Rovan, Peter Ruzicka (eds.), lecture notes in computer science, 841, google books, chapter 5 realizability models, p.93

Thomas Streicher, Realizability Models for $CZF + \neg Pow$, unpublished note. In this note WISC is called $TTCA_f$ ($TTCA$ stands for “type-theoretic collection axiom).

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