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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.
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. [1] The Dedekind number M ( n ) {\displaystyle M(n)} is the number of monotone Boolean functions of n {\displaystyle n} variables.
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 ...
A Dedekind domain can also be characterized in terms of homological algebra: an integral domain is a Dedekind domain if and only if it is a hereditary ring; that is, every submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective. [3]
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 ...
Chapters 1 to 4 cover similar ground to Gauss' Disquisitiones Arithmeticae, and Dedekind added footnotes which specifically cross-reference the relevant sections of the Disquisitiones. These chapters can be thought of as a summary of existing knowledge, although Dirichlet simplifies Gauss's presentation, and introduces his own proofs in some ...
The Dedekind-Kummer theorem holds more generally than in the situation of number fields: Let be a Dedekind domain contained in its quotient field , / a finite, separable field extension with = [] for a suitable generator and the integral closure of .
When interpreted as a proof within a first-order set theory, such as ZFC, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory ...