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Register a new userThe Ground Axiom (GA)
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Jonas Reitz
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A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent.
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A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing.
Menger conjectured that subsets of $mathbb R$ with the Menger property must be $sigma$-compact. While this is false when there is no restriction on the subsets of $mathbb R$, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We show that the perfect set version of the Open Graph Axiom for projective sets of reals, with consistency strength only an inaccessible cardinal, also implies Mengers conjecture restricted to this family of subsets of $mathbb R$.
This paper investigates Voevodskys univalence axiom in intensional Martin-Lof type theory. In particular, it looks at how univalence can be derived from simpler axioms. We first present some existing work, collected together from various published and unpublished sources; we then present a new decomposition of the univalence axiom into simpler axioms. We argue that these axioms are easier to verify in certain potential models of univalent type theory, particularly those models based on cubical sets. Finally we show how this decomposition is relevant to an open problem in type theory.
One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category $mathcal{E}$ can be (essentially uniquely) embedded into its exact hull ${mathcal{E}}^{textrm{ex}}$; this embedding induces a derived equivalence $textbf{D}^b(mathcal{E}) to textbf{D}^b({mathcal{E}}^{textrm{ex}})$. Whereas it is well known that Quillens obscure axioms are redundant for exact categories, some one-sided exact categories are known to not satisfy the corresponding obscure axiom. In fact, we show that the failure of the obscure axiom is controlled by the embedding of $mathcal{E}$ into its exact hull ${mathcal{E}}^{textrm{ex}}.$ In this paper, we introduce thr
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