cardinal number
Quick Summary
A cardinal number, in mathematics, represents the size of a set, indicating how many elements it contains. For instance, the number 3 is the cardinal number of any set with three elements.12
In the realm of infinite sets, cardinal numbers are used to compare their sizes. The smallest infinite cardinal number is aleph-null (\u2135\u2080), which is the cardinality of the set of natural numbers {1, 2, 3, ...}. Sets with this cardinality are called countably infinite. For example, the set of even numbers is countably infinite because its elements can be paired with the natural numbers.2
There are larger infinite cardinal numbers, such as the cardinality of the real numbers, denoted by c. The continuum hypothesis suggests that there are no cardinal numbers between aleph-null and the next cardinal number, aleph-one (\u2135\u2081). These infinite cardinal numbers are known as transfinite numbers.3
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metalogicSatisfaction of a theory by a structure: finite and infinite models…the concept of a “cardinal number,” which—for a finite set—is simply the number at which one stops in counting its elements. For infinite sets, however, the elements must be matched from set to set instead of being counted, and the “sizes” of these sets must thus be designated by…
Read More…the concept of a “cardinal number,” which—for a finite set—is simply the number at which one stops in counting its elements. For infinite sets, however, the elements must be matched from set to set instead of being counted, and the “sizes” of these sets must thus be designated by…
Read More- metalogicGeneralizations and extensions of the Löwenheim-Skolem theorem
…model, then, for any infinite cardinality α, that theory has a model of cardinality α. More explicitly, this theorem contains two parts: (1) If a theory has a model of infinite cardinality β, then, for each infinite cardinal α that is greater than β, the theory has a model of…
Read More…model, then, for any infinite cardinality α, that theory has a model of cardinality α. More explicitly, this theorem contains two parts: (1) If a theory has a model of infinite cardinality β, then, for each infinite cardinal α that is greater than β, the theory has a model of…
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- set theoryCardinality and transfinite numbers
) The application of the notion of equivalence to infinite sets was first systematically explored by Cantor. With ℕ defined as the set of natural numbers, Cantor’s initial significant finding was that the set of all rational numbers is equivalent to ℕ…
Read More) The application of the notion of equivalence to infinite sets was first systematically explored by Cantor. With ℕ defined as the set of natural numbers, Cantor’s initial significant finding was that the set of all rational numbers is equivalent to ℕ…
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infinityMathematical infinities…notion of its size, or cardinality. Unlike a finite set, an infinite set can have the same cardinality as a proper subset of itself. Cantor used a diagonal argument to show that the cardinality of any set must be less than the cardinality of its power set—i.e., the set that…
Read More…notion of its size, or cardinality. Unlike a finite set, an infinite set can have the same cardinality as a proper subset of itself. Cantor used a diagonal argument to show that the cardinality of any set must be less than the cardinality of its power set—i.e., the set that…
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