Definition:Finite
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Definition
Finite Set
A set $S$ is defined as finite iff $\exists n \in \N: S \sim \N_n$.
That is, if there exists an element $n$ of the set of natural numbers $\N$ such that the set of all elements of $\N$ less than $n$ is equivalent to $S$.
Finite Extended Real Number
An extended real number $x \in \overline{\R}$ is defined as finite iff:
- $\exists n \in \N: -n < x < n$
That is, an extended real number is finite iff there is a natural number which is greater, and its negative smaller, than it. This is the same as saying that $x$ is neither $+\infty$ nor $-\infty$, which in turn is the same as saying that $x$ is a regular real number (because any real number $x \in \R$ is finite in this definition).
Also see
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 13$: Arithmetic
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.1$
- Seth Warner: Modern Algebra (1965): $\S 17$
- A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 2.1$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 15$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 2 \ \text{(e)}$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 1.7$, Appendix $\text{A}.1, \ \text{A}.6$
