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Axiomatic Method

    The problem is to erect the entire structure of geometry upon the simplest foundation possible; i.e. to choose a minimum number of undefined elements and relations and a set of axioms concerning them, with the property that all of geometry can be logically deduced from these without further appeal to intuition.

Definition (Axiomatic Method) The following list is a technical description of what is meant by an axiomatic system.

    (i) Any axiomatic system must contain a set of technical terms that are deliberately chosen as undefined, called undefined terms, and are subject to the interpretation of the reader.
    
    (ii) All other technical terms of the system are ultimately defined by means of the undefined terms, and are called definitions.
    
    (iii) The axiomatic system contains a set of statements, dealing with undefined terms and definitions, that are chosen to remain unproved, and are called axioms.
    
    (iv) All other statements of the system must be logical consequences of the axioms. These derived statements are called the propositions of the axiomatic system.
    

    In an axiomatic system the proof of a specific statement is simply a sequence of statements, each of which follows logically from ones before and leads from a statement that is know to be true, to the statement that is to be proved; and so, in any axiomatic approach to a subject a complete understanding of the rules of logic is essential.

Logic Rule 1.  The following are types of justifications allowed for statements in proofs:

    (i) By hypothesis.
    
    (ii) By axiom.
    
    (iii) By a previously proven proposition.
    
    (iv) By definition.
    
    (v) By a previous established step in the same proof.
    
    (vi) By rule of logic (truth table).

Variables in mathematical statements can be quantified two different ways. First, the symbol is called a universal quantifier and is used to express that a variable may take on any value in a given collection. For example. can mean any of the following,
    
    (i) "For any "
    
    (ii) "For every "
    
    (iii) "For all "
    
    (iv) "Given any "
    
    (v) "If is any ..."

The other quantifier (called an existential quantifier) is used to express that a variable can take on at least one value in a given collection. For example, can mean any of the following,  

    (i) "For some "
    
    (ii) "There exists an "
    
    (iii) "There is an "
    
    (iv) "There are "
    

Mathematical statements are sometime made informally and you may have to rephrase a given statement before applying one of the following logic rules.

Logic Rule 2. The negation of the statement: " is the statement ".

Logic Rule 3. The negation of the statement: " is the statement   ".

The following logic rule is sometime called the method of exhaustion.

Logic Rule 4. (Rule of Elimination) Suppose that is a complete (finite) set of logical possibilities, that some (or none) of them imply and all the rest (at least one) imply Then is valid.

Often an outline (or column) proof is used when starting a new axiomatic system (or learning to write proofs). Next we describe the direct and indirect methods of proof. Notice that the direct proof is just a chain of implications, while the indirect proof makes use of logic rule 4.

Column Form of a Direct Proof Given the three previously proven propositions: Theorem 1: Theorem 2: Theorem 3: we can now prove the next theorem using a direct proof.  

    THEOREM: If then

    PROOF: We will use the direct proof method.

Column Form of an Indirect Proof  Given the three previously proven propositions: Theorem 1: Theorem 2: and Theorem 3: A complete set of logical possibilities is: and that is, one of the statements and are true and only one. We can now prove the next theorem using an indirect proof.

    THEOREM: If then

    PROOF: We will use the indirect proof method.


     

Example (Proof by Contradiction) Prove that the square root of 2 is an irrational number using an indirect proof.

The indirect proof techniques has two main variants: "proof by cases" and "proof by contradiction".

For readability, a proof usually starts with Proof, then declares what type of proof is involved (states a technique), and ends with some mark such as Q.E.D. or Since David Hilbert's times the Foundations of Geometry has served as a good starting point to learn how to write proofs.

Math Logic Books

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