Mathematics and Statistics Faculty and Staff Publications

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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


Xiquan, Phoenix



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neutrosophic logic, mathematics, neutrosophic set

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