Abstract
An algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of illoyal models: tail stretches, transposition twists, and maximal twists.
Highlights
The construction of algebra-valued models of set theory starts from an algebra A and a model of set theory forming an A-valued model of set theory
If the algebra A is a Boolean algebra, this construction results in Boolean-valued models of set theory which are closely connected to the theory of forcing and independence proofs in set theory [1]
How closely does the logic of the algebra-valued model of set theory resemble the logic of the algebra it is constructed from? In this paper, we introduce the concepts of loyalty and faithfulness to describe the relationship between the logic of the algebra A and the logical phenomena witnessed in the A-valued model of set theory: a model is called loyal to its algebra if the propositional logic in the model is the same as the logic of the algebra from which it was constructed and faithful if every element of the algebra is the truth value of a sentence in the model
Summary
The construction of algebra-valued models of set theory starts from an algebra A and a model of set theory forming an A-valued model of set theory. If the algebra A is a Boolean algebra, this construction results in Boolean-valued models of set theory which are closely connected to the theory of forcing and independence proofs in set theory [1]. If the algebra A is not a Boolean algebra, the construction gives rise to algebra-valued models of set theory whose logic is, in general, not classical logic. Examples of this are Heyting-valued models of intuitionistic set theory, lattice-valued models, orthomodular-valued models, and an algebra-valued model of paraconsistent set theory of Lowe and Tarafder [10, 25, 16, 14, 24].
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