Axioms and Ordinals, The First Three Axioms

The First Three Axioms

Here are the first three axioms in standard set theory.

Axiom 0, There is a Set

Formally, one might write this as:

some x { x = x }

Axiom 1: Extensionality

The members determine the set. If two sets have the same members they are the same set.

all x,y { all z { z ∈ x iff z ∈ y } then x = y }

Axiom 2: Comprehension

Any describable subcollection of a set is a set. This is actually an axiom scheme, rather than a single axiom. It applies to any formula in logic. In the following representation, f is any formula such that z is the only free variable.

all x, f { some y { all z { z ∈ x & f(z) iff z ∈ y }}}

If f had a free variable y, we might write f as z ∉ y. If x is nonempty, there is some y such that the members of x are in y iff they aren't in y. This is a contradiction. We wouldn't want set theory to become inconsistent after only 3 axioms, thus y cannot be a free variable in f.

At this point we have our first theorem, the existence of the empty set. Combine the set x in axiom 0 with the formula z ≠ Z using comprehension. No members satisfy this formula, hence there is a set y with no members. By extensionality the empty set, which is denoted ∅, is unique.

The next theorem runs Russell's paradox to show there is no universal set U that contains every set. Restrict U by the formula z ∉ z to get a set S whose members are all the sets that don't contain themselves. Therefore there is no set of all sets.

So far there is no reason to believe there is anything other than the empty set in our universe, so we need more axioms. The next 2 axioms create larger sets from existing sets, instead of restricting them to subsets.