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.