Definition:Strictly Positive/Integer
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Definition
The strictly positive integers are the set defined as:
- $\Z_{> 0} := \set {x \in \Z: x > 0}$
That is, all the integers that are strictly greater than zero:
- $\Z_{> 0} := \set {1, 2, 3, \ldots}$
Also known as
Some sources to not treat $0$ as a positive integer, and so refer to:
- $\Z_{> 0} := \set {1, 2, 3, \ldots}$
as the positive integers.
Consequently the term non-negative integers tends to be used in such sources for:
- $\Z_{\ge 0} := \set {0, 1, 2, 3, \ldots}$
Sources which are not concerned with the axiomatic foundation of mathematics frequently identify the positive integers with the natural numbers, which is usually completely appropriate.
Writers whose aim is specialised may refer to the positive integers as just numbers, on the grounds that these are the only type of number they are going to be discussing.
Also see
Sources
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.8$: Collections of Sets: Definition $8.4$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 4$: Number systems $\text{I}$: The rational integers
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $1$
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 1$: Introduction
- 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.1$ Divisibility of integers
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $1$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers
- 1989: George S. Boolos and Richard C. Jeffrey: Computability and Logic (3rd ed.) ... (previous) ... (next): $1$ Enumerability
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes
- 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 1$ What is infinity?