Hyperreal number

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In mathematics, the hyperreal numbers are the elements of one of several possible field extensions ∗ℝ of the field of real numbers ℝ, extensions that include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite when |x|<n for some integer n. Similarly, x is said to be infinitesimal when |x|<1/n for all positive integers n. The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about ℝ are also valid in ∗ℝ. For example, the commutative law of addition, x+y=y+x, holds for the hyperreals just as it does for the reals; since ℝ is a real closed field, so is ∗ℝ. Similarly, since sin⁡(πn)=0 for all integers n, one also has sin⁡(πh)=0 for all hyperintegers h. The transfer principle for ultrapowers is a consequence of Łoś's theorem of 1955. Concerns about the soundness...

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