Cantor–Bernstein–Schroeder theorem

In set theory, the set of Cantor - Bernstein -Schröder or short equivalence set (in the literature inconsistently as Cantor - Bernstein -Schröder shear [ equivalence ] set, set of Cantor - Bernstein equivalence theorem of Cantor - Bernstein, set of Schröder- Bernstein or the like hereinafter) an indication of the widths of two sets. Named after the mathematicians Georg Cantor, Felix Bernstein, and Ernst Schröder equivalence set (although also made ​​an exemplary contribution in the proof Richard Dedekind ) is an important tool in the detection of equal cardinality of two sets.

History

The equivalence theorem was formulated in 1887 by Georg Cantor, but not proved until 1897 by 19 - year-old Felix Bernstein in a seminar led by Georg Cantor and about the same time regardless of Ernst Schröder. Cantor told Bernstein's proof in the same year Émile Borel at First International Congress of Mathematicians in Zurich.

Cantor had first reported this equivalence set in his philosophical treatise on the theory of transfinite releases from 1887 ( without proof). In his great work contributions to the creation of transfinite set theory from 1895 Cantor has that sentence again drawn up and concluded from the Vergleichbarkeitssatz for cardinal numbers. However, the Vergleichbarkeitssatz could not prove Cantor. He is ( the problem of the well-ordering, 1915 About ) with the axiom of choice ( or selection principle or well-ordering theorem ) is equivalent by Friedrich Moritz Hartogs.

Dedekind himself found the proof of the equivalence theorem ( which was found in his estate ) on July 11 1887 but he did not publish it, and also did not share it with Cantor. Ernst Zermelo discovered Dedekind's proof again and gave in 1908 in his essay studies on the foundations of set theory I, a proof, where he What on Dedekind's chain theory of Dedekind's writing and what are the numbers? (1888 ) resorted.

Ernst Schröder 1896 ( About two definitions of finiteness and G. Cantor sets) published a proof sketch, however, turned out to be false, as Alwin Reinhold Korselt 1911 ( About a proof of the equivalence theorem ) had noticed; Schröder has there confirmed the flaw in his proof.

That the set without axiom of choice is provable, Richard Dedekind in 1887 and Amber in 1898 shown in his thesis ( Bernstein's proof first studies appeared in Borel's Leçons sur la théorie des fonctions and then again in Bernstein's memoir of set theory ).

A fitting name for the equivalence theorem would be Cantor - Dedekind equivalence set or Cantor -Dedekind Amber shear equivalence set. In addition, Bernstein has pointed out that Cantor himself had suggested the name " equivalence theorem ".

Set

The Cantor - Bernstein - Schroeder theorem is:

Two sets are called the same cardinality if there is a bijective mapping between them. In terms of the widths of the theorem and reads:

This exactly applies then, if and are equally powerful, and holds if and only if the same cardinality as a subset of, that is if there is an injective mapping from to. In terms of the properties of functions is the theorem:

Idea of ​​proof

Below here is a proof of concept is given.

Define the quantities:

Then set for each of:

Since in the case that it is not in, must be in, there is a uniquely determined element and is a well- defined mapping from to.

It can be shown that this function is the desired bijection.

Note that this definition is not constructive, that is, there is no method for arbitrary sets and injections, to decide in a finite number of steps whether a is made ​​in or not. For special quantities and pictures of course this can be possible.

A short and easily understandable proof can be found in the Goschen - ribbon set theory Erich Kamkes.

Illustration

Can be demonstrated to the definition of the basis of the adjacent display.

Shown are parts of the ( disjoint ) sets and as well as the illustrations and. Considering incorporated as a graph, then decomposes the graph into several connected components. These can be divided into four types:

( of each type is represented here one because the path through the element is to be on both sides infinite). However, it is generally not decidable in finitely many steps, which type has the current passing through a given element path.

The quantity defined in section proof idea now contains exactly the elements of which are part of an in incipient path. The mapping is defined so that it establishes a bijection of the elements on " adjacent in the path" elements within each connected component (this one has at the both sides infinite paths and the finite cycles, a choice of direction and you put on "backwards" fixed ).

Generalization

The Cantor - Bernstein - Schroeder theorem proves to be a direct consequence of Banach's mapping theorem.