In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A.
From Galileo's paradox, there exists a bijection that maps every natural number n to its square n2.
[2][1] There are definitions of finiteness and infiniteness of sets besides the one given by Dedekind that do not depend on the axiom of choice.
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.
Since A is infinite, the function "drop the last element" from B to itself is surjective but not injective, so B is dually Dedekind-infinite.
When sets have additional structures, both kinds of infiniteness can sometimes be proved equivalent over ZF.
The term is named after the German mathematician Richard Dedekind, who first explicitly introduced the definition.
Although such a definition was known to Bernard Bolzano, he was prevented from publishing his work in any but the most obscure journals by the terms of his political exile from the University of Prague in 1819.
In fact, the distinction was not really realised until after Ernst Zermelo formulated the AC explicitly.
With the general acceptance of the axiom of choice among the mathematical community, these issues relating to infinite and Dedekind-infinite sets have become less central to most mathematicians.
If we assume the axiom of countable choice (i. e., ACω), then it follows that every infinite set is Dedekind-infinite.
A von Neumann regular ring R has the analogous property in the category of (left or right) R-modules if and only if in R, xy = 1 implies yx = 1.