Results 261 to 270 of about 5,065 (309)
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2013
The ZF axioms allow us to assert the existence of any set whose members are selected according to some definable “rule”—this is essentially what the replacement schema says. However, we often want to assert the existence of a set without knowing a rule for selecting its members.
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The ZF axioms allow us to assert the existence of any set whose members are selected according to some definable “rule”—this is essentially what the replacement schema says. However, we often want to assert the existence of a set without knowing a rule for selecting its members.
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1974
For the deepest results about partially ordered sets we need a new settheoretic tool; we interrupt the development of the theory of order long enough to pick up that tool.
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For the deepest results about partially ordered sets we need a new settheoretic tool; we interrupt the development of the theory of order long enough to pick up that tool.
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2012
In 1904, Zermelo published his first proof that every set can be well-ordered. The proof is based on the so-called Axiom of Choice, denoted AC, which, in Zermelo’s words, states that the product of an infinite totality of sets, each containing at least one element, itself differs from zero (i.e., the empty set).
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In 1904, Zermelo published his first proof that every set can be well-ordered. The proof is based on the so-called Axiom of Choice, denoted AC, which, in Zermelo’s words, states that the product of an infinite totality of sets, each containing at least one element, itself differs from zero (i.e., the empty set).
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Compactness and the Axiom of Choice
Applied Categorical Structures, 1996Four versions of compactness (equivalent in ZFC) and their properties are investigated without the axiom of choice. All the four versions are equivalent iff the axiom of choice holds (many equivalent forms concerning mainly products of spaces are given).
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2002
The Axiom of Choice was first enunciated by Zermelo. The standard formulation is as follows.
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The Axiom of Choice was first enunciated by Zermelo. The standard formulation is as follows.
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1991
AbstractIn § 9.4 a principle — the axiom of countable choice — was introduced which differed from the axioms of this book's default theory because it asserted the existence of a set of a particular sort (actually, in this case, a sequence) without supplying a condition that characterizes it uniquely.
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AbstractIn § 9.4 a principle — the axiom of countable choice — was introduced which differed from the axioms of this book's default theory because it asserted the existence of a set of a particular sort (actually, in this case, a sequence) without supplying a condition that characterizes it uniquely.
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Cancer epigenetics in clinical practice
Ca-A Cancer Journal for Clinicians, 2023Veronica Davalos, Manel Esteller
exaly
2018
This is a mainly technical chapter concerning the causal embodiment of the Axiom of Choice from set theory. The Axiom of Choice powered a construction of an infinite fair lottery in Chapter 4 and a die-rolling strategy in Chapter 5. For those applications to work, there has to be a causally implementable (though perhaps not compatible with our laws of ...
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This is a mainly technical chapter concerning the causal embodiment of the Axiom of Choice from set theory. The Axiom of Choice powered a construction of an infinite fair lottery in Chapter 4 and a die-rolling strategy in Chapter 5. For those applications to work, there has to be a causally implementable (though perhaps not compatible with our laws of ...
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Understanding ligand-protected noble metal nanoclusters at work
Nature Reviews Materials, 2023María Francisca Matus, Hannu Häkkinen
exaly