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Aristotle – Cantor-Schröder-Bernstein Theorem

 Aristotle
Aristotle
(born 384 bce)   Greek philosopher and scientist who developed syllogistic logic as a formal scholastic discipline. He defined the syllogism as a “discourse in which certain things having been stated, something else follows of necessity from their being so.” Aristotle’s philosophy of the infinite, in which he held that only the potential infinite, and not any collection which was actually infinite, could be entertained by the reason, held great authority for 2,000 years, until Georg Cantor introduced his theory of transfinite sets in the 19th century. Aristotle wrote widely on every field of learning, interpreted and disagreed with the writings of his teacher Plato, and served as tutor to Alexander the Great.

arithmetic   The mathematical theory of addition, subtraction, multiplication, and division of integers. Formally, arithmetic is usually axiomatized by the so-called Peano axioms, and the theory is then often referred to as PA (for “Peano arithmetic”).

Aronszajn tree   Set Theory: For a a regular cardinal, an a-Aronszajn tree is a tree T with |T| = a, such that every chain in T is of cardinality less than a. Every Suslin tree is an Aronszajn tree. There are no Aronszajn trees of height w.

associative   An operation “ · ” on a set A is associative if for all a, b, and c in A, (a · b) · c = a · (b · c).
Cf. commutative, distributive.

associative property   A property of numbers which states that the operations of addition and multiplication are associative.
Cf. commutative, distributive.

asymmetric relation   A relation “~” on a set X is asymmetric if, for all x and y in X, x ~ y implies that it is not the case that y ~ x.
Cf. symmetric relation, antisymmetric relation.

automorphism   An isomorphism of a space to itself, that is, a bijective function on a space that preserves relationships, operations, and/or all essential properties (continuity, order, etc.).
Cf. homomorphism, group automorphism, ring automorphism, field automorphism.

automorphism group   The automorphisms of any object or structure can be composed to produce new automorphisms of that object. Since functional composition is associative, and since automorphisms are by their nature invertible, the automorphisms of an object or structure form a group (with the trivial automorphism serving as the identity element). This group is called the automorphism group of that object, usually denoted by Aut(X) for some object X. Automorphism groups are very useful, as they yield information about the symmetries of objects at hand.

axiom   In formal mathematics, a formula or schema of formulas stipulated as true in the theory under discussion, i.e., assumed to be true at the outset, and so not requiring proof. Axioms are the counterpart in mathematics of suppositions, assumptions, or premises in ordinary syllogistic logic.

axiom of choice   An axiom of set theory, particularly ZFC, which asserts that from any (infinite) family of sets a new set may be created containing exactly one element from each set in the family. This axiom has been shown to be independent of the other axioms of ZFC (ZF). It is equivalent to: 1) Zorn's Lemma; 2) the Hausdorff Maximality Theorem; 3) the well-ordering principle; and 4) the statement that the cartesian product of an infinite family of sets is non-empty. Because the axiom of choice permits non-constructive proofs, it is rejected by intuitionism, and careful mathematicians refer explicitly to its use in proofs which require it.

Banach-Tarski Paradox   Given two bounded subsets of 3-dimensional Euclidean space, provided each has interior points, then the first may be decomposed into finitely many pieces and translated by rigid motions into a subset congruent to the second subset. For instance, a ball of unit radius can be decomposed into finitely many pieces, and then reassembled into two balls of unit radius. This shocking result is a consequence of the axiom of choice.

base   Number systems: In a place-notational number system, the number of symbols used. For example, in base two the two symbols 0 and 1 are used, and in base seven the seven symbols 0, 1, 2, 3, 4, 5, and 6 are used.
Exponential expressions: The number or expression which is “being raised to” the power of the exponent.
Topology: Given a topological space X, a base is a class B of sets such that, for every x in X and every neighborhood U of x, there is a set b in B such that x is contained in b and b is contained in U.

bijection   A bijective function, i.e., a function that is both an injection and a surjection.

bijective   A function is bijective if it is both injective and surjective, i.e., both “one-to-one” and “onto.”

binary operation   A binary operation on a set X is a function whose domain is the set of ordered pairs of elements from X and whose range is X.
Examples: addition on the set of integers, function composition.

bound   A lower bound of any subset of a linear order (linearly ordered set) is an element which is less than or equal to every element of the subset. The greatest lower bound is the largest of its lower bounds. An upper bound of any subset of a linear order is an element which is greater than or equal to every element of the subset. The least upper bound is the smallest of the upper bounds.
Cf. infimum, supremum.

bounded   A set or sequence of values is called bounded if there is a value M such that the values are never bigger than M and never smaller than –M. A function is bounded if its values are bounded. A sequence of functions is pointwise bounded if at every domain element x the sequence of values of the functions at x is bounded. A subset E of a locally compact topological space is bounded if there exists a compact set C such that E is contained in C. Such a subset is called s-bounded if there exists a sequence of compact sets Ci such that E is contained in their union. See also: totally bounded.

Bourbaki, Nicolas   Although the myth of an actual French mathematician named Bourbaki was maintained for many years, it became generally known in 1952 that “Bourbaki” is the nom de plume of a private congress of young French mathematicians which formed in 1935, with the purpose of writing a comprehensive exposition of modern mathematics in many volumes, called Elements. Bourbaki’s original members included André Weil, Henri Cartan, Claude Chevalley, and Jean Dieudonné. Other mathematicians have been recruited into Bourbaki in the years since, including G. de Rham, A. Grothendieck, S. Lang, and R. Thom. A principle motivation for the activity of the Bourbaki group is the felt need to unify the entire corpus of mathematical work in the face of accelerating diversification and specialization of mathematics.

 Georg Cantor
Cantor, Georg
(born 1845)   German mathematician and founder of modern set theory. Cantor’s concern with the set-theoretic nature of the real numbers began while he was working on certain properties of trigonometric series during the early 1870’s. Inspired by the work of his friend Dedekind, Cantor proved in 1873 that the rational numbers are countable, and later the same year proved that the real numbers themselves are uncountable. These proofs introduced important methods, including the notion of a one-to-one correspondence and the method of diagonalization. Cantor’s results were counterintuitive and broke with established dogma about the nature of infinity. Consequently, he became a controversial figure, opposed by some mathematicians (e.g., Kronecker) who used their influence to stifle Cantor’s career as much as possible. This opposition exacerbated Cantor’s predisposition to severe depression, and he ultimately died, of a heart attack, in a sanitorium.

Related article: Cantor's Theorem
Related article: Cantor Set
Related MiniText: Infinity -- You Can't Get There From Here...

Cantor-Bendixson Theorem   Every closed subset of the real numbers is the disjoint union of a perfect set and a countable set.
Cf. derived set.

Cantor-Schröder-Bernstein Theorem   For any sets A and B, if there exists an injective (one-to-one) function from A into B and also an injective function from B into A, then there exists a surjective (onto) function from A onto B.

Aristotle – Cantor-Schröder-Bernstein Theorem

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