Real Numbers and Logic
The real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics.
Abraham Robinson's theory of nonstandard or hyperreal numbers extends the set of the real numbers by infinitesimal numbers, which allows building infinitesimal calculus in a way closer to the usual intuition of the notion of limit. Edward Nelson's internal set theory is a non-Zermelo–Fraenkel set theory that considers non-standard real numbers as elements of the set of the reals (and not of an extension of it, as in Robinson's theory).
The continuum hypothesis posits that the cardinality of the set of the real numbers is, i.e. the smallest infinite cardinal number after, the cardinality of the integers. Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is, one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.
Read more about this topic: Real Number
Famous quotes containing the words real, numbers and/or logic:
“Surrealism, n. Pure psychic automatism, by which it is intended to express, whether verbally or in writing, or in any other way, the real process of thought. Thought’s dictation, free from any control by the reason, independent of any aesthetic or moral preoccupation.”
—André Breton (1896–1966)
“Our religion vulgarly stands on numbers of believers. Whenever the appeal is made—no matter how indirectly—to numbers, proclamation is then and there made, that religion is not. He that finds God a sweet, enveloping presence, who shall dare to come in?”
—Ralph Waldo Emerson (1803–1882)
“The much vaunted male logic isn’t logical, because they display prejudices—against half the human race—that are considered prejudices according to any dictionary definition.”
—Eva Figes (b. 1932)