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The Axiomatic Method

# The Axiomatic Method - The Axiomatic Method The Axiomatic...

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The Axiomatic Method The Axiomatic Method is a procedure, which involves a chain of propositions and their proofs, to establish the correctness of principles which had been formulated by experiment, by trial and error, or by intuitive insight. This chain of statements produced by the Axiomatic Method is called an Axiomatic System. An Axiomatic System has Four Components 1. A set of undefined terms 2. A set of defined terms 3. A set of AXIOMS (also called Postulates) 4. A set of THEOREMS together with their proofs (See Table 1.2.1 (p. 9) for a description of these components.) The proven statements are called the theorems of the axiomatic system. A statement cannot be called a theorem until a valid proof of that statement has been developed. A statement which has been proposed as being possibly true (but which has not yet been proved) is called a conjecture or a hypothesis . A theorem whose sole purpose is to be used in the proof of another theorem is called a Lemma . A theorem whose proof is little more than the simple application of a previously proved theorem is called a Corollary of the previously proved theorem.

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2 Example of an Axiom System: The Fe-Fo System Undefined terms : Fe’s, Fo’s, and the relation belongs to . Defined terms : When a Fe belongs to a Fo, that Fo is said to contain that Fe. Axioms : Axiom 1. There exist exactly three distinct Fe’s in this system. Axiom 2. Any two distinct Fe’s belong to exactly one Fo. Axiom 3. Not all Fe’s belong to the same Fo. Axiom 4. Any two distinct Fo’s contain at least one Fe that belongs to both. Some Theorems
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The Axiomatic Method - The Axiomatic Method The Axiomatic...

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