In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets.
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Anonymous2013-08-31 23:29
Exponentiation is non-decreasing in both arguments:
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Anonymous2013-09-01 0:14
Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of the philosophical and mathematical schools of constructivism and intuitionism. Typical dumb goy.
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Anonymous2013-09-01 0:59
In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only.
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Anonymous2013-09-01 1:45
A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.
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Anonymous2013-09-01 2:30
The difficulty appears when there is no natural choice of elements from each set. If we cannot make explicit choices, how do we know that our set exists? For example, suppose that X is the set of all non-empty subsets of the real numbers.
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Anonymous2013-09-01 3:15
Tychonoff's theorem stating that every product of compact topological spaces is compact.