نتایج جستجو برای: axiom of choice
تعداد نتایج: 21175756 فیلتر نتایج به سال:
We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a ⊆-maximal subfamily with the finite intersection property and the principle asserting that if φ is a property of finite character then every set has a ⊆-maximal subset of which φ...
1950 the 18-year old Edward Nelson posed the problem of finding χ (see its history in [S]). A number of relevant results were obtained under additional restrictions on monochromatic sets. K. Falconer, for example, showed [F] that χ is at least 5 if monochromatic sets are Lebesgue measurable. Amazingly though, the problem has withstood all assaults in general case, leaving us with embarrassingly...
The Boolean prime ideal theorem is a very slightly weaker version of the axiom of choice. In 2011, Andreas Blass showed that models of set theory (with atoms) in which the Boolean prime ideal theorem holds but the full axiom of choice fails correspond with a certain type of topological group. The path taken to prove this surprising result draws not just from set theory but also from combinatori...
What becomes of constructive constructive mathematics without the axiom of (countable) choice? Using illustrations from a variety of areas, it is argued that it becomes better. Despite the apparent unanimity among schools of constructive mathematics with respect to the acceptance of the countable axiom of choice, I believe it to be one of the central problems, or mysteries, of constructive math...
The assumption of [HM] and our assumption here, the existence of a strong partition cardinal, is moderately special. On the one hand, it violates the Axiom of Choice and is not relatively consistent with ZF (unlike AC and its negation). On the other hand, under the Axiom of Determinacy (AD), such cardinals are abundant and consistent with countable choice and DC, the principle of Dependent Choi...
Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations to establish consistency, independence and conservativity results. van Oosten’s contribution to the Workshop (see van Oos...
We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice. §1. The problem. It is well-known that combinatory logic is inconsistent with the choice schema ACV for operations: (∀x)(∃y)A(x, y)→ (∃f)(∀x)A(x, fx) (1) where A is an arbitr...
Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, e...
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