Cantor theorem

The set $2^A$ of all subsets of a set $A$ is not equipotent to $A$ or to any subset of it. The idea behind the proof of this theorem, due to G. Cantor (1878), is called "Cantor's diagonal process" and plays a significant role in set theory (and elsewhere). Cantor's theorem implies that no two of the sets

$$2^A,2^{2^A},2^{2^{2^A}},\dots,$$

are equipotent. In this way one obtains infinitely many distinct cardinal numbers (cf. Cardinal number). Cantor's theorem also implies that the set of all sets does not exist. This means that one must not include among the axioms of set theory the assertion that for each propositional function (or predicate) $\phi(x)$ there exists a set consisting of all elements $x$ satisfying $\phi(x)$ (see [1], [2], [3], [8]).

B.A. Efimov

Any decreasing sequence of non-empty bounded closed sets of real numbers has a non-empty intersection. This generalizes to compact subsets of a metric space. The property: If the diameters of a decreasing sequence of non-empty closed sets in a metric space $X$ tend to zero then the sets have non-empty intersection, is one of the definitions of completeness of $X$. The property that every totally ordered decreasing family of non-empty closed sets of a topological space $X$ has a non-empty intersection, is one of the definitions of compactness of $X$ (see [1], [2], [4], [5], [11]).

Every set of real numbers is the union of the perfect set of its condensation points (cf. Condensation point of a set) and a countable set. This sometimes called the Cantor–Bendixson theorem. It generalizes to the case of a subset of a metric space with a countable basis (Lindelöf theorem, see [1], [2], [3], [14], [15], [17]).

If each of two sets is equipotent to a subset of the other, then these two sets are equipotent. A similar statement holds for two well-ordered sets. This is sometimes called the Cantor–Bernstein theorem or simply Bernstein's theorem (F. Bernshtein gave a correct proof of this theorem, see [1], [2], [3], [10], [12], [16]).

If

$$A_n=a_n\cos nx+b_n\sin nx\to0$$

for all but a finite number of points of interval $[-\pi,\pi]$, then $a_n,b_n\to0$. This generalizes to the case when $A_n\to0$ on a set of positive measure (Lebesgue's theorem), when $A_n\to0$ on a set of the second category (Young's theorem), as well as to other situations. Important corollaries are various theorems on sets of uniqueness of trigonometric series (see [1], [6], [7], [9], [18], [19]).

A continuous function on a closed bounded interval of the real line is uniformly continuous on it. This generalizes to continuous mappings from a compact space to a uniform space. This is sometimes called the Heine–Cantor theorem (see [1], [4], [5], [13]).

References

[1]	G. Cantor, E. Zermelo (ed.) , Gesammelte Abhandlungen , Springer (1932)
[2]	F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))
[3]	K. Kuratowski, A. Mostowski, "Set theory" , North-Holland (1968)
[4]	P.S. Aleksandrov, "Einführung in die Mengenlehre und in die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[5]	N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
[6]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[7]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
[8]	G. Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre" J. Reine Angew. Math. , 84 (1878) pp. 242–258
[9]	G. Cantor, "Über trigonometrische Reihen" Math. Ann. , 4 (1871) pp. 139–143
[10]	G. Cantor, "Beiträge zur Begründung der transfiniten Mengenlehre" Math. Ann. , 49 (1897) pp. 207–246
[11]	G. Cantor, "Über unendliche lineare Punktmannigfaltigkeiten" Math. Ann. , 17 (1880) pp. 355–358
[12]	E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1928)
[13]	E. Heine, "Die Elemente der Funktionenlehre" J. Reine Angew. Math. , 74 (1872) pp. 172–188
[14]	E. Lindelöf, "Remarque sur une théorème fondamental de la théorie des ensembles" Acta Math. , 29 (1905) pp. 183–190
[15a]	G. Cantor, "Über einen die trigonometrischen Reihen betreffenden Lehrsatz" J. Reine Angew. Math. , 72 (1870) pp. 130–138
[15b]	G. Cantor, "Beweis, dass eine für jeden reellen Wert von $x$ durch eine trigonometrische Reihe gegebene Funktion $f(x)$ sich nur auf eine einzige Weise in dieser Form darstellen lässt" J. Reine Angew. Math. , 72 (1870) pp. 139–142
[16]	G. Cantor, "Über unendliche lineare Punktmannigfaltigkeiten" Math. Ann. , 21 (1883) pp. 51–58
[17]	L. Bendixson, "Quelques théorèmes de la théorie des ensembles de points" Acta Math. , 2 (1883) pp. 415–429
[18]	H. Lebesgue, "Leçons sur les séries trigonométriques" , Gauthier-Villars (1906)
[19]	W.H. Young, Proc. Roy. Soc. , 87 (1912) pp. 331–339
[20]	N. Bourbaki, "Eléments d'histoire des mathématiques" , Hermann (1960)

M.I. Voitsekhovskii

Comments

1) Two sets are equipotent (equivalent) if there is a bijection (a one-to-one onto mapping) between them. Intuitively this means that they have the same number of elements; in other words: They have (or are of) the same cardinality. Cantor's proof is as follows: Assume $f\colon A\to2^A$ is a mapping; to show that it is not onto, consider $X=\lbrace a\in A\colon a\notin f(a)\rbrace$. Then $X$ is not in the range of $f$.

In an analogous way he showed that the set of real numbers is not countable: Assume $f\colon\mathbf N\to\lbrace x\colon0<x<1\rbrace$ is a mapping and write $f(n)=0.a_{n1}a_{n2}\dots$ in decimal expansion. Define $r=0.r_1r_2\dots$ by $r_i=5$ if $a_{ii}\neq5$ and $r_i=6$ if $a_{ii}=5$. Then $r$ is not in the range of $f$. This last proof best explains the name "diagonalization process" or "diagonal argument".

4) This theorem is also called the Schroeder–Bernstein theorem. A similar statement does not hold for totally ordered sets, consider $\lbrace x\colon0<x<1\rbrace$ and $\lbrace x\colon0<x\leq1\rbrace$.

For general historical information see [20]. For set of the second category cf. Category of a set.