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The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2 μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces , though they lack some of the properties that logarithms of positive real numbers possess.
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that κ {\displaystyle \kappa } is a regular cardinal if and only if every unbounded subset C ⊆ κ {\displaystyle C\subseteq \kappa } has cardinality κ {\displaystyle \kappa } .
The aleph numbers differ from the infinity commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the ...
The smallest infinite cardinal number is . The second smallest is ℵ 1 {\displaystyle \aleph _{1}} ( aleph-one ). The continuum hypothesis , which asserts that there are no sets whose cardinality is strictly between ℵ 0 {\displaystyle \aleph _{0}} and c {\displaystyle {\mathfrak {c}}} , means that c = ℵ 1 {\displaystyle {\mathfrak {c ...
There are many other famous integer sequences, such as the sequence of Fibonacci numbers, the sequence of factorials, ... Cardinal numbers: ...
The first two in the sequence are by far the most common; 'tertiary' appears occasionally, ... For other numbers, the elements of the cardinal number are used, ...
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 ...
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written ,,,, …, where is the Hebrew letter beth.