How To Use Cardinality In A Sentence
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Indeed, because cardinality is permutation-invariant, every cardinality quantifier is included, including “there are infinitely many”, “there are uncountably many”, and others that are not first-order definable.
Logical Constants
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Models of different cardinality obviously cannot be isomorphic, hence any theory, complete or incomplete, which has at least one model of infinite cardinality, will have a multiverse associated with.
Archive 2009-06-01
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For example, it certainly depends on whether your set of trials is countably infinite or uncountably infinite (in other words the cardinality of your set of trials).
A Short Critique of Bradley Monton's Paper
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We understand that sets have a cardinality, that is, that collections have a number associated with them and it doesn't really matter what the members of that set are.
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Dedekind also provided a proof of the Cantor-Bernstein Theorem (that between any two sets which can be embedded into each other one-to-one there exists a bijection, so that they have the same cardinality), another basic result in the modern theory of transfinite cardinals.
Dedekind's Contributions to the Foundations of Mathematics
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For example, it certainly depends on whether your set of trials is countably infinite or uncountably infinite (in other words the cardinality of your set of trials).
A Short Critique of Bradley Monton's Paper
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One boy, Joshua, used a pointer to illustrate a math concept known as cardinality, by completing place settings on a whiteboard.
KansasCity.com: Front Page
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(In the case of a complete theory, the models of different cardinality will be elementarily equivalent, even if they are non-isomorphic).
Archive 2009-06-01
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(Newman 1928, 144) To see how this so-called cardinality constraint applies to ramseyfications of theories, note that in Carnap's hands, the non-observational part of reconstructed theories, their theoretical entities, were represented by “purely logico-mathematical entities, e.g. natural numbers, classes of such, classes of classes, etc.”
Vienna Circle
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As for the aleph-null and aleph-one: it was proven that the continuum hypothesis essentially whether the cardinality of real numbers is aleph-one or higher is undecidable in standard set theory, so whether you want to accept it or not, you won’t hit any contradictions.
Intelligent Design explained: Part 2 random search - The Panda's Thumb
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Both sets have a count or cardinality equal to the transfinite number א 0 - aleph-null.
TierneyLab
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For example, it certainly depends on whether your set of trials is countably infinite or uncountably infinite (in other words the cardinality of your set of trials).
A Short Critique of Bradley Monton's Paper
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Given that we have an expert in the house, we consulted Math Man at dinner, who told us that we have Cantor to thank for the solution to this problem of the cardinality or countability of the integers and other infinite sets of numbers.
Archive 2007-03-01
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Hausdorff proved further results on the cardinality of Borel sets in 1916.
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Bonnay (2008) argues for a different criterion, invariance under potential isomorphism, which counts finite cardinality quantifiers and the notion of finiteness as logical, while excluding the higher cardinality quantifiers ” thus “[setting] the boundary between logic and mathematics somewhere between arithmetic and set theory”
Logical Constants
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A definition of the fuzzy degree of digital image is presented by introducing some concepts in fuzzy mathematics, such as triangular norm operator and non fuzziness cardinality.
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Frege's approach to providing a logical analysis of cardinality, the natural numbers, infinity and mathematical induction were groundbreaking, and have had a lasting importance within mathematical logic.
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Thus, if you write card (N) = µ 0 (read: aleph sub zero), his theorems justified calling the cardinality of the “second number class” µ 1.
The Early Development of Set Theory
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At recess, one boy, Joshua, used a pointer to illustrate a math concept known as cardinality, by completing place settings on a whiteboard.
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