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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 number
A natural number n whose distinct divisors, including 1 but not including n itself, sum to n.
Example: 6 is perfect since its divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. The first three perfect numbers are 6, 28, and 496. Any number of the form 2n – 1(2n – 1) is perfect provided that 2n – 1 is a Mersenne prime. Such numbers are called Euclid numbers. Euler proved that all even perfect numbers are Euclid numbers. It is not known whether there are infinitely many perfect numbers, or if there are any odd perfect numbers.
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.
poset
A partially ordered set, i.e., a set with a partial order defined on it.
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.
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.
prime number
Any natural number greater than 1 that is evenly divisible only by itself and 1. There are infinitely many prime numbers. The number of primes less than a given number n is denoted p(n), and approaches the value n/lnn for sufficiently large n.
Related article: Fundamental Theorem of Arithmetic
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.
proper factor
See factor.
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.
Pythagorean triple
An ordered triple (a,b,c) of natural numbers satisfying a2 + b2 = c2. The triples (3,4,5) and (5,12,13) are the first of infinitely many examples.
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 k ln 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.
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