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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). 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.