Дедекинд шексіздігі және жиындар теориясы

Set with an equinumerous proper subset

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. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind finite if it is not Dedekind infinite (i. e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. A simple example is , the set of natural numbers. From Galileo's paradox, there exists a bijection that maps every natural number n to its square n2. Since the set of squares is a proper subset of , is Dedekind infinite. Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory, most mathematicians assumed that a set is infinite if and only if it is Dedekind infinite. In the early twentieth century, Zermelo–Fraenkel set theory, today the most commonly used form of axiomatic set theory, was proposed as an axiomatic system to formulate a theory of sets free of paradoxes such as Russell's paradox. Using the axioms of Zermelo–Fraenkel set theory with the originally highly controversial axiom of choice included (ZFC) one can show that a set is Dedekind finite if and only if it is finite in the usual sense. However, there exists a model of Zermelo–Fraenkel set theory without the axiom of choice (ZF) in which there exists an infinite, Dedekind finite set, showing that the axioms of ZF are not strong enough to prove that every set that is Dedekind finite is finite. There are definitions of finiteness and infiniteness of sets besides the one given by Dedekind that do not depend on the axiom of choice. A vaguely related notion is that of a Dedekind finite ring.

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 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 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" that 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 that 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.

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 countably infinite 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 axiom CC (i. e., 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 (assuming consistency of ZF).

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, 1=xy = 1 implies 1=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.