author = "Gutman A.E.",

title = "Cumulative structure of a Boolean-valued model of set theory",

booktitle = "International conference on Geometric Analysis in honor of the 90th anniversary of academician Yu.G.Reshetnyak (Novosibirsk, September, 22--28, 2019): Proceedings",

address = "Novosibirsk",

publisher = "Sobolev Institute of Mathematics SB RAS",

year = "2019",

isbn = "978-5-4437-0949-9",

pages = "64--66",

annote = "We show that every Bolean-valued universe has a multilevel structure analogous to the von Neumann cumulative hierarchy, in which, at each level, the ascents are added of the Boolean-valued functions defined on subsets of the previous levels. Another cumulative structure is obtained if we consider the ascents of constant functions only and add mixings at the limit steps. Such cumulative hierarchies clarify the structure of Boolean-valued systems and, in particular, make it possible to easily prove the uniqueness of a Boolean-valued universe up to isomorphism. We also present a general tool for adding ascents to Boolean-valued systems which builds the cumulative hierarchy starting from an arbitrary extensional system. This makes it possible to construct examples of Boolean-valued systems with unusual properties. By means of the tool, we show that each of the five conditions listed in the definition of a Boolean-valued universe, is essential and does not follow from the other conditions."