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The axioms of ZF

1.  
2. Axiom of extensibility
   
   Sets are uniquely determined by their members. There cannot be two different sets that have the same members.

    

3. Axiom of the empty set
   ]
   There is an empty set that contains no set.

    

4. Axiom of unordered pairs
   
   For any two sets there is a third set that contains those two sets and only those two sets.

    

5. Axiom of union
   
   For every set x there is a set y that contains the union of the sets that are members x. In other words for every set x there is a set y such that t is a member of y if and only if there is some z that is a member of x and t is a member of z

    

6. Axiom of infinity
   
   There is a set that contains the empty set and if any set y is in then the set containing the union of y and the set containing y is also in . By induction contains every finite integer.

    

7. Axiom of replacement
   
   

   where
   This is an infinite set of axioms, one for each statement in the language of ZF. We must expand to and surround the statement with universal quantifiers on `'. We can enumerate all the but we cannot enumerate all the sets that can be specified as fixed parameters in a statement. Thus we cannot index all statements with all parameters with the integers. We must use quantifiers ranging over all sets to include every possible statement in the language of ZF with every possible value for a fixed parameter.

   The axiom of replacement allows us to make a new set v from any statement in the language of ZF (with any fixed parameters) that defines y uniquely as a function of x and any set u. A set r belongs to v if and only if there is a set s that belongs to u and r is the image of s under the function defined by .

    

8. Axiom of the power set
   
   For any set x there is a set y that includes every subset of x. This is the axiom that defines sets of higher cardinality or at least seems to. All the subsets of a set definable in a given formal system does not include all possible subsets. In fact the sets definable in a formal system are recursively enumerable and thus countable no matter how pretentious the axioms of the system are.

The important thing to understand about the axioms is that they are comparatively simple precise rules for deducing new statements from existing ones. One can easily write a computer program that will implement these rules and print out all the statements that one can prove from these axioms. Of course some mathematicians think there is a Platonic heaven of all integers, all subsets of the integers and all true sets. They see these axioms as more than a formal system. They see the axioms as telling us about this idealized mathematical truth.

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