On the Structure of the Boolean-Valued Universe

Gutman, A. E.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 2.

Abstract:
The logical machinery is clarified which justifies declaration of hypotheses. In particular, attention is paid to hypotheses and conclusions constituted by infinitely many formulas. The formal definitions are presented for a Boolean-valued algebraic system and model of a theory, for the system of terms of the Boolean-valued truth value of formulas, for ascent and mixing. Logical interrelations are described between the ascent, mixing, and maximum principles. It is shown that every mixing with arbitrary weights can be transformed into a mixing with constant weight. The notion of restriction of an element of a Boolean-valued algebraic system is introduced and studied. It is proven that every Boolean-valued model of Set theory which meets the ascent principle has some multilevel structure analogous to von Neumann's cumulative hierarchy.

Keywords: set theory, Boolean-valued model, universe, cumulative hierarchy

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