# Set Theory Axioms
Specifically, ZFC is a collection of approximately 9 axioms (depending on convention and precise formulation) that, taken together, define the core of mathematics through the usage of set theory. More formally, ZFC is a predicate logic equipped with a binary relation ∈, which refers to set membership and is read as "in". To be clear, it is said that a∈b when a is an element of b.
In general, statements in set theory are expressed using first-order logic, which uses a number of quantifiers (or logical symbols):
means "is in", as in the introduction. means "for all"; e.g. (translated: for all real ) is a way of expressing the trivial inequality. means "there exists"; e.g. (translated: for all real , there exists a real such that y^3=x) is a way of stating that every real number has a real cube root. means "is equivalent to". For example, is a way of expressing the fact that is positive if and only if is positive. means "implies". For example, s a way of expressing the fact that the square of a positive number is positive. Note that , since is a false statement (e.g. for ). means "logical and"; e.g. is a way of expressing the fact that is positive and is negative; i.e. is negative.
This allows for the axioms in ZFC to be stated succinctly using symbols, as in the following section.
# Formal Definition (Axioms)
The axioms of ZFC can be stated in several equivalent ways, and have slightly different names and logical formulations depending on the source. Of course, each individual source will have a rigorous correct treatment of the axioms, one of which follows:
# Axiom of extensionality
In other words, if
# Axiom of pairing
In other words, for all
# Axiom of comprehension
Axiom of comprehension: the elements of
In plain language, this statement is equivalent to "Given any property
# Axiom of union
In other words, for all
# Axiom of power set
In other words, for any set
# Axiom of infinity
In simpler terms, an infinite set exists.
# Axiom of replacement
A function takes any set
# Axiom of regularity
In other words, for all non-empty sets
- No set can be an element of itself. This resolves Russell's paradox.
- Every set has a smallest element with respect to ∈.
These 8 axioms define a consistent theory, ZF (though, of course, it is very difficult to prove that this system is consistent). When the axiom of choice is added to the eight axioms above, the theory becomes ZFC (the "C" for choice), and it is this system that is commonly used as the foundation of mathematics.