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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. [1] A simple example is , the set of natural numbers.
When S is finite, its completion is also finite, and has the smallest number of elements among all finite complete lattices containing S. [ 12 ] The partially ordered set S is join-dense and meet-dense in the Dedekind–MacNeille completion; that is, every element of the completion is a join of some set of elements of S , and is also the meet ...
The ring = of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal I of R, R/I is a finite set, and recall that a finite integral domain is a field; so by (DD4) R is a Dedekind domain. As above, this includes all the examples considered ...
For example, {,,,,} is a finite set with five elements. The number of ... Using the standard ZFC axioms for set theory, every Dedekind-finite set is also finite, ...
The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. [2] It can be used to prove many of the fundamental results of real analysis , such as the intermediate value theorem , the Bolzano–Weierstrass theorem , the extreme value theorem , and the Heine ...
Axiom 4, which requires the order to be Dedekind-complete, implies the Archimedean property. The axiom is crucial in the characterization of the reals. For example, the totally ordered field of the rational numbers Q satisfies the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first ...
However, over a Dedekind domain the ideal class group is the only obstruction, and the structure theorem generalizes to finitely generated modules over a Dedekind domain with minor modifications. There is still a unique torsion part, with a torsionfree complement (unique up to isomorphism), but a torsionfree module over a Dedekind domain is no ...
For example, the set of even natural numbers is equinumerous to the set of all natural numbers. A set that is equinumerous to a proper subset of itself is called Dedekind-infinite. [1] [3] The axiom of countable choice (AC ω), a weak variant of the axiom of choice (AC), is needed to show that a set that is not Dedekind-infinite is actually finite.