Generic filters in partially ordered sets

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1982
Authors
Eslami, Esfandiar
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Mathematics
Abstract

The concept of partial-order valued models and that of D-generic filters play a central role in the present day development of Set Theory;In this dissertation, we consider questions related to both of the above concepts;In Section 2, we prove the existence of a model (M,(ELEM)) for the unrestricted Comprehension Scheme;((FOR ALL)x)((x (epsilon) M) (--->) ((THERE EXISTS)s)((s(ELEM)M) (WEDGE) ((VBAR)(VBAR)x (ELEM) s(VBAR)(VBAR) = (VBAR)(VBAR)F(x)(VBAR)(VBAR))));in a certain n-valued logic (L(,n), N, D) with connectives N and D defined by appropriate truth tables;In Section 3, we construct partial-order valued models where (VBAR)(VBAR)x (epsilon) y(VBAR)(VBAR) is a subset of a partially ordered set P with (VBAR)(VBAR)(IL-PERP)F(VBAR)(VBAR) defined as z (VBAR)z (epsilon) P and z is incompatible with every element of (VBAR)(VBAR)F(VBAR)(VBAR) and with the other connectives defined in their usual set-theoretical interpretations. These models are reduced to two valued models via the notion of a generic filter of P. In Section 4, we introduce the notion of a molecule m of P by requiring that every two elements compatible with m be themselves compatible. Then, we prove the equivalence of the existence of a generic filter to that of a molecule in a partially ordered set. In Section 5, we introduce some P-lattice algebras and prove the existence of a D-complete ultrafilter in a Boolean algebra for the denumerable case;In Section 6, we introduce the notion of k-inducive partially ordered sets as partially ordered sets in which every inversely well-ordered subset of cardinality less then k of nonzero elements has a nonzero lower bound. Based on k-inducive partially ordered set, we prove the existence of some E-complete filters of partially ordered sets with the condition E(' )<(' )k imposed on cardinality of E;In Sections 7 to 10, we prove some equivalent forms of Martin's Axiom in connection with Boolean algebras and also we give some consequences of Martin's Axiom pertaining to the cardinal exponentiation;Finally, in Section 11 we introduce the notion of a receding sequence (S(,i))(,i

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Fri Jan 01 00:00:00 UTC 1982