Statement
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f(s) is an element of s. With this concept, the axiom can be stated:
- For any set X of nonempty sets, there exists a choice function f defined on X.
Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.
Each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a same set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family. The axiom of choice asserts the existence of such elements; it is therefore equivalent to:
- Given any family of nonempty sets, their Cartesian product is a nonempty set.
Read more about this topic: Axiom Of Choice
Famous quotes containing the word statement:
“The force of truth that a statement imparts, then, its prominence among the hordes of recorded observations that I may optionally apply to my own life, depends, in addition to the sense that it is argumentatively defensible, on the sense that someone like me, and someone I like, whose voice is audible and who is at least notionally in the same room with me, does or can possibly hold it to be compellingly true.”
—Nicholson Baker (b. 1957)
“Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.”
—Charles Sanders Peirce (1839–1914)
“One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in one’s theme, it needs no translation into art, the bold statement of it being sufficient.”
—Rebecca West (1892–1983)