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A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph-null, the smallest infinite cardinal. In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set.
The continuum hypothesis says that =, i.e. is the smallest cardinal number bigger than , i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC , a standard axiomatization of set theory; that is, it is impossible to prove the continuum ...
To define ℵ α for arbitrary ordinal number α, we must define the successor cardinal operation, which assigns to any cardinal number ρ the next larger well-ordered cardinal ρ + (if the axiom of choice holds, this is the (unique) next larger cardinal). We can then define the aleph numbers as follows: ℵ 0 = ω ℵ α+1 = (ℵ α) +
In linguistics, and more precisely in traditional grammar, a cardinal numeral (or cardinal number word) is a part of speech used to count. Examples in English are the words one , two , three , and the compounds three hundred [and] forty-two and nine hundred [and] sixty .
Cardinal functions are widely used in topology as a tool for describing various topological properties. [2] [3] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [4] prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the ...
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For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with n {\displaystyle n} elements has a total of 2 n {\displaystyle 2^{n}} subsets, and the theorem holds because 2 n > n {\displaystyle 2^{n}>n} for all non-negative integers .
Numbers were the symbols of the divine order of the universe. “However, the Greek mathematician and mystic Pythagoras studied these great teachings and formed a more cohesive philosophy of numbers.