Consistency and Completeness - Consistency, Independence,...

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Consistency, Independence, and Completeness of a Set of Axioms Recall that an axiomatic system has four components 1. A set of undefined terms 2. A set of defined terms 3. A set of AXIOMS (or Postulates) 4. A set of THEOREMS together with their proofs EXAMPLE of an Axiomatic System: Four-Point Geometry Undefined Terms: Point, line, on Axioms: Ax 1: There exist exactly four points. Ax 2: (Any) Two distinct points are on exactly one line. Ax 3: Each line is on exactly two points. Definition: A MODEL for an axiomatic system is a structure (in a “real world” context or in an abstract context) and an interpretation (from within that structure) of the undefined terms such that all the Axioms become TRUE statements about the structure.
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Definition: A set of Axioms is Inconsistent if it is possible to prove one statement and its negation as both being TRUE. A set of Axioms is Consistent if it is Not Inconsistent. The set of axioms in an axiomatic system is proven to be
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This note was uploaded on 11/02/2010 for the course MATH modern geo taught by Professor Shirley during the Spring '10 term at University of Texas at Austin.

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Consistency and Completeness - Consistency, Independence,...

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