Zermelo Fraenkel Axioms

We have read everything in mathematics can be derived from these.

Before embarking on this adventure to visit all the axioms, we need to know about the “everything is a set” constraint:

In ZFC, everything is a set. If you see A={b,c,d}, we are looking at four sets, not just one. If you are not a set, the bouncer won’t let you into the ZFC night club.

Integers are sets:

  • 0 = {} = \varnothing
  • 1 = {0}
  • 2 = {0,1}
  • and so on…

Notice that 0 has no elements, 1 has one element, 2 has two elements, etc.

We changed our format after we noticed that different authors were writing different things. We will include brief mentions of axioms below, and place them in alphabetically order. We will try to have individual pages for most of them, with those pages showing more explanation, and possibly even mention of controversy/dissent and our own ongoing questions.

Choice

For any set S of pairwise disjoint, nonempty sets, there exists a set that contains one element from each set in S.

Pairwise disjoint means that for any two distinct sets in S, there are no elements in the intersection between the sets. We say distinct so someone won’t try using the same set twice to make the pair.

Elementary Sets

  • A set exists with nothing in it and is called by at least two names, “null set” and “empty set”.  The symbol \ is used.
  • A set exists with a single element and is called a “singleton”.
  • For any two objects ‘a’ and ‘b’, a set {a,b} may be constructed and this set is an Unordered Pair

Existence

\exists x (x=x)

A set exists and it is equal to itself.

Extensionality

If two sets have the same elements then they are equal.

Power Set

For any set S, the collection of all subsets of S into a set is called the Power Set of S.

Separation

If a property P is well formed then for any set S, a set may be constructed consisting of all the elements of S for which the property P is true.

Example, let Property P be, “an integer is even” and let S={1,2,3,4,5}. The constructed set would be {2,4}.

Specification, Schema of

same as Separation

Union

For any set S, the collection of the members of the members of S is also a set, and is called the Union.

Given a first set A and a second set B which is a subset of A, any elements x of the universal set X, which are elements of B, must also be elements of A.