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In order to study the axiomatization of the construct over possibly non-halting programs and tests, this paper introduces the notion of -sets by considering the tests from an abstract -algebra. When the -algebra is an ada, the axiomatization is shown to be complete by obtaining a subdirect representation of -sets. Further, this paper considers the equality test with the construct and gives a complete axiomatization through the notion of agreeable -sets. [ABSTRACT FROM AUTHOR]

A ternary relation ${\hbox to 0pt{\hss\mid$\hss}\limits_\hbox to 0pt{\hss\smile$\hss}}$ between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be closely related to modular pairs in the lattice of algebraically closed sets. [ABSTRACT FROM AUTHOR]

In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2ω = ω2 and that $L({\mathscr{P}}(\omega_1))$ satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □(κ) fails for all regular κ > ω1. [ABSTRACT FROM AUTHOR]