rently no single philosophy that truly defines our mathematical and logic foundations,
especially when it comes to combining both mathematical knowledge and the applica-
tion of mathematics to physical reality.
AXIOMATIC SYSTEM
What are axiomsand postulates?
These two words are often treated as the same; in fact, some mathematicians consider
the word axioma slightly archaic synonym for postulate. Although both are considered
to be a proposition (statement) that is true without proof, there are subtle differences.
An axiom in mathematics refers to a general statement that is true without proof,
and it is often related to equality, such as “two things equal to the same thing are
equal to each other,” and those related to operations. They should also be consistent—
it should not be possible to deduce any contradictory statements from the axiom.
A postulate is also a proposition (statement) that is true without proof, but it deals
with a specific subject matter, such as the properties of geometric figures. Thus, it is
not as general as an axiom. For example, Euclidean geometry is based on the five pos-
tulates known, of course, as Euclid’s postulates. (See below; for more about Euclid, see
“History of Mathematics” and “Geometry and Trigonometry.”)
What is an axiomatic system?
An axiomatic system is a logical system that has a definite set of axioms; from these
axioms, theorems can be derived. In each system, propositions (statements) are
proved on the basis of a limited number of axioms or postulates—all with a few unde-
fined terms. The other terms are defined on the basis of the undefined terms. One of
the first axiomatic systems was Euclidean geometry.
Overall, an axiomatic system has several basic components: the undefined terms
of the system (primitives); well-formed formulas, or how symbols are put into the sys-
tem based on certain allowed rules, sometimes called defined terms; axioms, or what
is also known as “self-evident truths” of the system; theorems, or statements that are
proved based on axioms or other proven theorems; and, finally, the rules of inference,
or those that allow moves from certain formulas to other formulas.
How are some parts of an axiomatic systemfurther defined?
There are several terms that further define an axiomatic system. All of them are slight-
ly intertwined, depending on the system.
The absence of contradiction—or the ability to prove a proposition (statement)
and its negative are both true—is known as consistency.Independence is not neces- 115