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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Example of an Axiom System:
The FeFo 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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 Spring '10
 Shirley
 Euclidean geometry, Axiom, fo, Axiomatic system

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