Stone Duality

THIS IS NATURE SIGNALING THE END OF DUALITY 
THIS IS HUGE - CHANGE THE ENTIRETY OF CREATION - NO DUALITY ANYWHERE.
IT IS VERY CLEARLY A PREDETERMINED ADVANCEMENT OF HUMAN CONSCIOUSNESS - THIS BUBBLE IS TOO WELL DESIGNED - EUPHORIA CANCELS KARMA

We study various generalizations and weakenings of the Rasiowa-Sikorski Lemma for Boolean algebras. Building on previous work from Goldblatt, we extend the Rasiowa-Sikorski Lemma to co-Heyting algebras and modal algebras, and show how this yields completeness results for the corresponding non-classical first-order logics. Moreover, working without the full power of the Axiom of Choice, we generalize the framework of possibility semantics from Humberstone, and more recently Holliday, in order to provide choice-free representation theorems for distributive lattice, Heyting algebras and co-Heyting algebras. We also generalize a weaker version of the Rasiowa-Sikorski Lemma for Boolean algebras, known as Tarski's Lemma, to distributive lattices, HA's and co-HA's, and use these results to define a new semantics for first-order intuitionisitic logic.

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone's representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone's representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomor-phisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.