From: tangent60@aol.com (Tangent60) Newsgroups: sci.math Subject: Re: Where are all the Axioms Date: 25 Aug 1998 20:47:21 GMT Klaus D. Witzel wrote: >Can someone point me to all the axioms, please. Here are all the axioms in ZFC, one of the most popular modern mathematical theories. Notation: A = for all = universal quantifier E = there exists = existential quantifier & = and V = or ~ = not => = implies <=> = equivalence e = element of (primitive) 0 = empty set (primitive) Capital letters denote sets, and objects are sets iff there are elements in them or they are the empty set. Here are the axioms. Axiom of Exstensionality: Two sets are equal iff they have the same elements. S = T <=> (Ax: x e S <=> x e T) Axiom of Union: Given any set of sets, there is a set which is the union of all the sets in the set. EU: Ax: x e U <=>(ES: x e S & S e T) Axiom of Power Set: Given any set, there is a set of all the subsets of the set. ET: Ax: x e T <=> (Ay: y e x => y e T) Axiom of Infinity: The set of natural numbers exists. EN: 0 e N & (x e N => x U {x} e N) Axiom of Regularity: Given any nonempty set, there is an element of the set whose intersection with the set is empty. ~S = 0 => Ex: x e S & Ay: y e x => ~y e S Axiom Schema of Replacement: Given any set S, if we can associate with each object x of S another object y uniquely, then the set of all such corresponding objects of elements in S exists. (In other words,) If p is a two-place predicate such that p(x, y) & p(x, z) => y = z, then EB: Ay: y e B <=> (Ex: x e A & p(x, y)) And last but definitely not least, Axiom of Choice: Given any set S of sets, there is a function F: S -> US such that F(X) e X for each X e S. (This takes a while to write out in predicate logic..) There are at most minor errors in these axioms.