follows: First, define a function over the natural numbers (that is, over the finite ordinals) , so that for every natural number n, f(n) is the set of finite subsets of X of size n (i.e. that have a bijection with the finite ordinal n). f(n) is never empty, or otherwise X would be finite (as can be proven by induction on n). The image of f is the countable set whose members are themselves infinite (and possibly uncountable) sets. By using the axiom of countable choice we may choose one member from each of these sets, and this member is itself a finite subset of X. More precisely, according to the axiom of countable choice, a (countable) set exists, so that for every natural number n, g(n) is a member of f(n) and is therefore a finite subset of X of size n. Now, we define U as the union of the members of G. U is an infinite countable subset of X, and a bijection from the natural numbers to U, , can be easily defined. We may now define a bijection that takes every member not in U to itself, and takes h(n) for every natural number to . Hence, X is Dedekind-infinite, and we are done. Generalizations Expressed in category-theoretical terms, a set A is Dedekind-finite if in the category of sets, every monomorphism is an isomorphism. 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, implies . More generally, a Dedekind-finite ring is any ring that satisfies the latter condition. Beware that a ring may be Dedekind-finite even if its underlying set is Dedekind-infinite, e.g. the integers. Notes References Faith, Carl Clifton. Mathematical surveys and monographs. Volume 65. American Mathematical Society. 2nd ed. AMS Bookstore, 2004. Moore, Gregory H., Zermelo's Axiom of Choice, Springer-Verlag, 1982 (out-of-print), , in particular pp. 22-30 and tables 1 and 2 on p. 322-323 Jech, Thomas J., The Axiom of Choice, Dover Publications, 2008, Lam, Tsit-Yuen. A first course in noncommutative rings. Volume 131 of Graduate Texts in Mathematics. 2nd ed. Springer, 2001. Herrlich, Horst, Axiom of Choice, Springer-Verlag, 2006, Lecture Notes in Mathematics 1876, ISSN print edition 0075–8434, ISSN electronic edition: 1617-9692, in particular Section 4.1. Basic concepts in infinite set theory Cardinal numbers de:Dedekind-unendlich