Comparison With The Usual Definition of Infinite Set
This definition of "infinite set" should be compared with the usual definition: a set A is infinite when it cannot be put in bijection with a finite ordinal, namely a set of the form {0,1,2,...,n−1} for some natural number n – an infinite set is one that is literally "not finite", in the sense of bijection.
During the latter half of the 19th century, most mathematicians simply assumed that a set is infinite if and only if it is Dedekind-infinite. However, this equivalence cannot be proved with the axioms of Zermelo–Fraenkel set theory without the axiom of choice (AC) (usually denoted "ZF"). The full strength of AC is not needed to prove the equivalence; in fact, the equivalence of the two definitions is strictly weaker than the axiom of countable choice (CC). (See the references below.)
Read more about this topic: Dedekind-infinite Set
Famous quotes containing the words comparison with the, comparison with, comparison, usual, definition, infinite and/or set:
“[Girls] study under the paralyzing idea that their acquirements cannot be brought into practical use. They may subserve the purposes of promoting individual domestic pleasure and social enjoyment in conversation, but what are they in comparison with the grand stimulation of independence and self- reliance, of the capability of contributing to the comfort and happiness of those whom they love as their own souls?”
—Sarah M. Grimke (1792–1873)
“What is man in nature? A nothing in comparison with the infinite, an all in comparison with the nothing—a mean between nothing and everything.”
—Blaise Pascal (1623–1662)
“The comparison between Coleridge and Johnson is obvious in so far as each held sway chiefly by the power of his tongue. The difference between their methods is so marked that it is tempting, but also unnecessary, to judge one to be inferior to the other. Johnson was robust, combative, and concrete; Coleridge was the opposite. The contrast was perhaps in his mind when he said of Johnson: “his bow-wow manner must have had a good deal to do with the effect produced.””
—Virginia Woolf (1882–1941)
“The treatment of the incident of the assault upon the sailors of the Baltimore is so conciliatory and friendly that I am of the opinion that there is a good prospect that the differences growing out of that serious affair can now be adjusted upon terms satisfactory to this Government by the usual methods and without special powers from Congress.”
—Benjamin Harrison (1833–1901)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
““. . . Me miserable! which way shall I fly
Infinite wrath and infinite despair?
Which way I fly is Hell; myself am Hell;
And, in the lowest deep, a lower deep
Still threat’ning to devour me opens wide,
To which the Hell I suffer seems a Heaven.”
—John Milton (1608–1674)
“Speak of me as I am. Nothing extenuate,
Nor set down aught in malice.”
—William Shakespeare (1564–1616)