Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics

Authors

Florentin Smarandache, University of New MexicoFollow

Document Type

Book

Publication Date

2002

Abstract

In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, that rigorously defines the infinitesimals. Informally, an infinitesimal is an infinitely small number. Formally, x is said to be infinitesimal if and only if for all positive integers n one has xxx < 1/n. Let &>0 be a such infinitesimal number. The hyper-real number set is an extension of the real number set, which includes classes of infinite numbers and classes of infinitesimal numbers. Let’s consider the non-standard finite numbers 1+ = 1+&, where “1” is its standard part and “&” its non-standard part, and –0 = 0-&, where “0” is its standard part and “&” its non-standard part. Then, we call ]-0, 1+[ a non-standard unit interval. Obviously, 0 and 1, and analogously nonstandard numbers infinitely small but less than 0 or infinitely small but greater than 1, belong to the non-standard unit interval. Actually, by “-a” one signifies a monad, i.e. a set of hyper-real numbers in non-standard analysis: .(-a)= {a-x: x%‘*, x is infinitesimal}, and similarly “b+” is a monad: .(b+)= {b+x: x%‘*, x is infinitesimal}. Generally, the left and right borders of a non-standard interval ]-0, 1+[ are vague, imprecise, themselves being non-standard (sub)sets .(-a) and .(b+) as defined above. Combining the two before mentioned definitions one gets, what we would call, a binad of “-c+”: .(-c+)= {c-x: x%‘*, x is infinitesimal} 4 {c+x: x%‘*, x is infinitesimal}, which is a collection of open punctured neighborhoods (balls) of c. Of course, –a < a and b+ > b. No order between –c+ and c. Addition of non-standard finite numbers with themselves or with real numbers: -a + b = -(a + b) a + b+ = (a + b)+ -a + b+ = -(a + b)+ -a + -b = -(a + b) (the left monads absorb themselves) a+ + b+ = (a + b)+ (analogously, the right monads absorb themselves) Similarly for subtraction, multiplication, division, roots, and powers of non-standard finite numbers with themselves or with real numbers. By extension let inf ]-0, 1+[ = -a and sup ]-0, 1+[ = b

Publisher

Xiquan, Phoenix

ISSN

1-931233-67-5

Language (ISO)

English

Keywords

neutrosophic logic, mathematics, neutrosophic set

Recommended Citation

Florentin Smarandache (ed.) Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics. Phoenix: Xiquan, 2002

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