Download The Non-Euclidean Revolution Material Axiomatic Systems and the

The Non-Euclidean Revolution
Material Axiomatic Systems and the Turtle Club Example
Recall that a material axiomatic system consists of four parts: the primitive (or undefined) terms,
the defined terms, the axioms (or assumptions) used as the starting point for deduction, and the
theorems (or statements requiring proof).
A proof is an “absolutely convincing argument” for the truth of a statement. It consists of a list of
statements, each having a justification, culminating in the desired conclusion. Only seven types of
justifications are allowed:
- by hypothesis (the hypothesis, or "given", of the theorem)
- by RAA hypothesis/conclusion (for proofs by contradiction)
- by Axiom ... (an axiom of the system)
- by Theorem ... (a previously-proven theorem)
- by Definition ... (the meaning of a defined term)
- by step ... (an earlier step in the proof)
- by rule ... of logic (a rule of logic)
----------------------------------------------------------------The Turtle Club example:
Primitive terms: person, collection -- these are used in their everyday senses.
Definitions: The Turtle Club is a collection of one or more persons. A person in the Turtle Club is
called a Turtle. The Committees are certain collections of one or more Turtles. A Turtle in a
Committee is called a member of that Committee. Two Committees are equal if they have identical
membership lists (not identical functions!). Two Committees having no members in common are
called disjoint Committees.
Axioms:
1. Every Turtle is a member of at least one Committee.
2. For every pair of Turtles there is one and only one Committee of which both are members.
3. For every Committee there is one and only one disjoint Committee.
Theorems:
1. Every Turtle is a member of at least two distinct Committees.
2. Every Committee has at least two members.
3. ??