# Intuitionistic logic model theory and forcing

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## Abstract

The independence proofs of Cohen for the axiom of choice, the continuum hypothesis, and the axiom of constructability are re-formulated using s. Kripke's intuitionistic logic model theory·. We define transfinite sequences of intuitionistic.models with a 'class' model limit in a manner exactly analogous to the definition of Godel in the classical case of a transfinite sequence of (domains of) classical models, M∞ , with a 'class' model limit, L. Classical independence results are established by working with the intuitionistic models themselves; no classical models are constructed, no countable classical models are required (though the definition of intuitionistic model is essentially the same as that of forcing.) ¶ An intuitionistic (or forcing) generalization of the R∞: sequence (sets with rank) is defined and some connections between it and Scott and Solovay's boolean valued models for set theory are established. ¶ For completeness sake, the first six chapters provide a complete treatment of s. Krlpke's intuitionistic logic model theory. Completeness proofs are given for tableau and axiomatic systems, compactness and Skolem-Lowenheim theorems ·are established, and relations with classical logic are shown. The connection between Kripke rmodel theory and algebraic model theory is shown in the propositional case.