BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry		Peano axioms – Ramsey’s Theorem Peano axioms   The axiom system developed by Giuseppe Peano to formalize arithmetic. The key to his method is the introduction of a “successor operation” S on numbers; if a is a number, then Sa is the successor of that number. In this way Peano reduced arithmetic to the conceptually primitive operation of counting. The axioms are: 0 is a number. If a is a number, then Sa is a number. If a and b are numbers, then a + 0 = a, and a + Sb = S(a + b). If a and b are numbers, then a × 0 = 0, and a × Sb = a × b + a. (Induction Principle.) For any formula f, if we have f(0) and if f(a) always implies f(Sa), then we have f(a) for all numbers a. perfect set   A closed set X is called perfect if every point of X is an accumulation point of X. The following are equivalent characterizations: X is closed and containes no isolated points. X is closed and dense in itself. X is equal to its derived set. pointwise bounded   See bounded. poset   A partially ordered set, i.e., a set with a partial order defined on it. positive set   Given a signed measure m on a measure space X, a measurable set A in X is called a positive set if the measure of all measurable subsets of A is greater than or equal to zero. Cf. negative set, null set. postulate   A statement in a mathematical theory that is assumed without proof. Essentially synonymous with “axiom.” potential infinite   A distinction made by Aristotle: a set is potentially infinite if it cannot be finitely completed, e.g., our naive or given conception of the natural numbers. Aristotle admitted potentially infinite sets, but denied the logical possiblity of the actual infinite, that is, infinite totalities considered as completed entities. Related MiniText: Infinity -- You Can't Get There From Here... power set   Given a set X, the power set of X, denoted P(X), is the collection of all subsets of X. If X is a finite set with n elements, then P(X) is a finite set with 2n elements. Cf. power set axiom. power set axiom   An axiom of set theory which states that, for any given set X, the power set, i.e., the collection of all subsets of X, exists and is a set. precompact   Given a topological space X, a subset E of X is called precompact if its closure is compact. predecessor   In a structure with an order relation defined upon it, the predecessor of an element b is the greatest element less than b. Cf. successor predicate calculus   A system of symbolic logic which augments the propositional calculus with quantification over variables. The two forms of quantification are existential and universal, and are denoted by respectively. This permits the construction of sentences such as which could be read, “There exists an x such that for all y, x times y is equal to y.” (Such a sentence would be true in arithmetic or group theory, for instance.) When quantification is permitted only over variables, the logic is first-order. If quantification is permitted over classes of variables or over predicates, the logic is second-order. proper class   A collection of elements that is not a set. For example, the collection of all sets must be thought of as a proper class in order to avoid the Russell paradox. propositional calculus   The formal system of symbolic logic in which sentences are treated as objects related by the logical connectives: The last two symbols are called the conditional and the biconditional respectively, but they are not essential; indeed all the connectives are fully expressible by the use of the two connectives for “or” and “not.” For example, “p implies q” may be equivalently expressed as “not p or q.” In the notation of propositional calculus, this equivalence may be written, Cf. predicate calculus. p-series   An infinite series of the form with p a positive real number. See the related article for details. Related article: Series quantifier   See predicate calculus. quasi-disjoint family   Set Theory: See D-system. Quine atom   In set theory, a set whose only member is itself, i.e., x = { x }. More generally, some phrases may be “Quined” to form meaningful sentences, e.g., “is a five-word phrase” is a five-word phrase. Ramsey cardinal   A cardinal k is called a Ramsey cardinal if Cf. Ramsey’s Theorem. Ramsey number   See Ramsey’s Theorem. Ramsey’s Theorem   Graph theory: given any two positive integers m and n, there exists a number, called the Ramsey number of m and n, and denoted R(m, n), such that any simple graph with R(m, n) vertices either contains a clique with m vertices (all vertices adjacent) or an independent set with n vertices (no vertices adjacent). The question of how to determine the Ramsey number for arbitrary m and n is unsolved and believed to be very difficult, but various lower and upper bounds on it have been proven over the years. In particular, it is known that R(m, m) 2m/2, and that R(m, n) (m + n - 2)C(m - 1). Also, certain specific Ramsey numbers are known, e.g., R(3, 3) = 6. Thus, in any group of six people, either three are mutual acquaintances or three are mutual strangers. Set theory: Let n be a positive integer, k and l be (finite or transfinite) cardinals, and denote by kln the assertion that the complete graph of size k in which each edge has been “colored” by one of n colors contains a complete subgraph of size l all of whose edges are the same color. More generally still, denote by the assertion that if the subsets of k of size b have each been colored with one of a-many colors, then there is a subset of X of size l all of whose b-sized subsets are the same color (homogeneous). Ramsey’s Theorem (infinite case) may now be stated as: For all positive integers k and l, we have i.e., for every coloring of the k-sized subsets of the natural numbers (w) into l colors, there is an infinite homogeneous set (an infinite collection of k-sized subsets all the same color). This more general notion gives rise to a field of set theory known as Ramsey Theory, in which far more is known about infinite sets than finite ones.
		Peano axioms – Ramsey’s Theorem

HOME | ABOUT | CONTACT | AD INFO | PRIVACY Copyright © 1997-2013, Math Academy Online™ / Platonic Realms™. Except where otherwise prohibited, material on this site may be printed for personal classroom use without permission by students and instructors for non-profit, educational purposes only. All other reproduction in whole or in part, including electronic reproduction or redistribution, for any purpose, except by express written agreement is strictly prohibited. Please send comments, corrections, and enquiries using our contact page.