In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite.
A vaguely related notion is that of a Dedekind-finite ring. A ring is said to be a Dedekind-finite ring if ab=1 implies ba=1 for any two ring elements a and b. These rings have also been called directly finite rings.
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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.)
The following conditions are equivalent in ZF. In particular, note that all these conditions can be proved to be equivalent without using the AC.
Every Dedekind-infinite set A also satisfies the following condition:
This is sometimes written as "A is dually Dedekind-infinite". It is not provable (in ZF without the AC) that dual Dedekind-infinity implies that A is Dedekind-infinite. (For example, if B is an infinite but Dedekind-finite set, and A is the set of finite one-to-one sequences from B, then "drop the last element" is a surjective but not injective function from A to A, yet A is Dedekind finite.)
It can be proved in ZF that every dually Dedekind infinite set satisfies the following (equivalent) conditions:
(Sets satisfying these properties are sometimes called weakly Dedekind infinite.)
It can be shown in ZF that weakly Dedekind infinite sets are infinite.
ZF also shows that every well-ordered infinite set is Dedekind infinite.
Since every infinite, well-ordered set is Dedekind-infinite, and since the AC is equivalent to the well-ordering theorem stating that every set can be well-ordered, clearly the general AC implies that every infinite set is Dedekind-infinite. However, the equivalence of the two definitions is much weaker than the full strength of AC.
In particular, there exists a model of ZF in which there exists an infinite set with no denumerable subset. Hence, in this model, there exists an infinite, Dedekind-finite set. By the above, such a set cannot be well-ordered in this model.
If we assume the CC (ACω), then it follows that every infinite set is Dedekind-infinite. However, the equivalence of these two definitions is in fact strictly weaker than even the CC. Explicitly, there exists a model of ZF in which every infinite set is Dedekind-infinite, yet the CC fails.
The term is named after the German mathematician Richard Dedekind, who first explicitly introduced the definition. It is notable that this definition was the first definition of "infinite" which did not rely on the definition of the natural numbers (unless one follows Poincaré and regards the notion of number as prior to even the notion of set). 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. Moreover, Bolzano's definition was more accurately a relation which held between two infinite sets, rather than a definition of an infinite set per se.
For a long time, many mathematicians did not even entertain the thought that there might be a distinction between the notions of infinite set and Dedekind-infinite set. In fact, the distinction was not really realised until after Ernst Zermelo formulated the AC explicitly. The existence of infinite, Dedekind-finite sets was studied by Bertrand Russell and Alfred North Whitehead in 1912; these sets were at first called mediate cardinals or Dedekind cardinals.
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. However, the study of Dedekind-infinite sets played an important role in the attempt to clarify the boundary between the finite and the infinite, and also an important role in the history of the AC.
Expressed in category-theoretical terms, a set A is Dedekind-finite if in the category of sets, every monomorphism f: A → A 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, xy = 1 implies yx = 1. 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.