Mohammad Golshani; Omid Etesami; Shahram Mohsenipour
Abstract
Cohen’s method of forcing is one of the main tools in set theory for constructing models of ZFC. In this paper, we consider different methods of introducing forcing, and show that they are all equivalent. First we introduce the method of forcing using partial orders and state some of its basic ...
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Cohen’s method of forcing is one of the main tools in set theory for constructing models of ZFC. In this paper, we consider different methods of introducing forcing, and show that they are all equivalent. First we introduce the method of forcing using partial orders and state some of its basic properties. Then we consider the method of Boolean-valued models and show that it is equivalent to the first approach using partial orders. We do this by showing that each forcing notion can densely be embedded into a complete Boolean algebra. Then we introduce the topological approach to forcing and compare it with the partial order approach to forcing. We show that the forcing relation defined in a topological manner is the same as the forcing relation defined using partial orders and hence these two methods are essentially identical. Finally we consider the categorical approach to forcing and compare it with the method of Boolean-valued models. We show that for a given complete Boolean algebra, the category of sheaves over it is essentially the same as the Boolean-valued model constructed using that Boolean algebra.
