Constructing illoyal algebra-valued models of set theory. (English) Zbl 1535.03254
Summary: 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.
MSC:
| 03E70 | Nonclassical and second-order set theories |
| 03E40 | Other aspects of forcing and Boolean-valued models |
| 03B55 | Intermediate logics |
| 03B20 | Subsystems of classical logic (including intuitionistic logic) |
| 03C90 | Nonclassical models (Boolean-valued, sheaf, etc.) |
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