In mathematics a formal system

is a set of axiom s and the rules of
logic for deriving theorems from those axioms. It can be thought of
as a computer program for outputting theorems. We can write a
program to output all the theorems for *any* formal system.
The axioms say what primitive objects and relationships exist and
how new objects can be constructed.

In set theory there is one primitive object, the empty
set , and one relationship, set
membership . All of mathematics can be
modeled with these primitives. For example the integer 1 is defined
as the set that contains the empty set. The integer two is the set
containing 1 and the empty set. Integer **N** is the set
containing the empty set and all integers less than **N**.

The most powerful generally accepted formal system is Zermelo
Fraenkel (ZF )
set theory . We will list the axioms of ZF
adapted from Cohen(50)[11]. First
we need to explain the notation. Sometimes we refer to arbitrary
statements in the language of ZF with upper case letters. **A**
refers to any valid statement in the language of ZF. is a
statement with a single parameter. is a statement with **k** parameters.

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