Dedekind-infinite set

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a set A is Dedekind-infinite (named after the German mathematician <a href="/facts/Richard_Dedekind/CLkWDhY3">Richard Dedekind</a>) if some proper <a href="/facts/Subset/SJGERmbA">subset</a> B of A is <a href="/facts/Equinumerous/7hMvasXx">equinumerous</a> to A. Explicitly, this means that there exists a <a href="/facts/Bijective_function/j4gTuTmW">bijective function</a> 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 <a href="/facts/Natural_number/u65og8JE">natural numbers</a>.
A simple example is 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
, the set of <a href="/facts/Natural_number/u65og8JE">natural numbers</a>. From <a href="/facts/Galileo%2527s_paradox/W81SgIRj">Galileo's paradox</a>, there exists a bijection that maps every natural number n to its <a href="/facts/Square_number/jfakASPK">square</a> n2. Since the set of squares is a proper subset of 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
, 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
 is Dedekind-infinite. 
Until the <a href="/facts/Foundational_crisis_of_mathematics/yjpbbUkt">foundational crisis of mathematics</a> showed the need for a more careful treatment of set theory, most mathematicians <a href="/facts/Tacit_assumption/rMjEC2GW">assumed</a> that a set is <a href="/facts/Infinite_set/9rTN0xGH">infinite</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it is Dedekind-infinite. In the early twentieth century, <a href="/facts/Zermelo%25E2%2580%2593Fraenkel_set_theory/UYIsPbXw">Zermelo–Fraenkel set theory</a>, today the most commonly used form of <a href="/facts/Axiomatic_set_theory/1ptfsvUP">axiomatic set theory</a>, was proposed as an <a href="/facts/Axiomatic_system/dwMz70d5">axiomatic system</a> to formulate a <a href="/facts/Theory_of_sets/1ptfsvUP">theory of sets</a> free of paradoxes such as <a href="/facts/Russell%2527s_paradox/XupqWGYt">Russell's paradox</a>. Using the axioms of Zermelo–Fraenkel set theory with the originally highly controversial <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a> included (ZFC) one can show that a set is Dedekind-finite if and only if it is <a href="/facts/Finite_set/za1newlP">finite</a> 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 <a href="/facts/Finite_set/za1newlP">definitions of finiteness and infiniteness of sets</a> besides the one given by Dedekind that do not depend on the axiom of choice.
A vaguely related notion is that of a <a href="/facts/Dedekind-finite_ring/a4R2LYea">Dedekind-finite ring</a>.

Dedekind-infinite set open-in-new

Dedekind-infinite set