An Infinitary Extension of Jankov’s Theorem

Studia Logica 86 (1):111-131 (2007)
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Abstract

It is known that for any subdirectly irreducible finite Heyting algebra A and any Heyting algebra, B, A is embeddable into a quotient algebra of B, if and only if Jankov's formula ${\rm{\chi A}}$ A for A is refuted in B. In this paper, we present an infinitary extension of the above theorem given by Jankov. More precisely, for any cardinal number ${\rm{\kappa }}$, we present Jankov's theorem for homomorphisms preserving infinite meets and joins, a class of subdirectly irreducible complete ${\rm{\kappa }}$ - Heyting algebras and ${\rm{\kappa }}$ - infinitary logic, where a ${\rm{\kappa }}$ -Heyting algebra is a Heyting algebra A with ${\rm{\# A}} \le {\rm{\kappa }}$ and ${\rm{\kappa }}$ - infinitary logic is the infinitary logic such that for any set ${\rm{\Theta }}$ of formulas with ${\rm{\# \Theta }} \le {\rm{\kappa }}$ ⌈ Θ and ‷ Θ are well defined formulas.

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A logic stronger than intuitionism.Sabine Görnemann - 1971 - Journal of Symbolic Logic 36 (2):249-261.
Kripke Completeness of Infinitary Predicate Multimodal Logics.Yoshihito Tanaka - 1999 - Notre Dame Journal of Formal Logic 40 (3):326-340.

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