sary to an axiomatic system, but consistency is definitely necessary. The opposite of
consistency in an axiomatic system is inconsistency.An axiomatic system is called independentif no other axioms can be derived (or
proved) from other axioms in the system; in other words, the entire axiomatic system
will be termed independent if all of its underlying axioms are independent. The inde-
pendence of a system is usually determined after the consistency. An axiomatic system
that is dependenthas some axioms that are redundant; this is also called redundancy.
An axiomatic system is completeif no additional axiom can be added to the system
without making the new system dependent or inconsistent. In other words, the aim is
to prove or disprove any statement about the objects in the system from the axioms
alone. In complete systems, every true proposition about the defined and undefined
terms can be proved from the axioms. Systems with the logic based on true or false
propositions connected by “and,” “or,” and “not” are complete, as are those that include
quantifiers. More complex systems, such as set theory, are not considered complete.What is an undefined term?
In terms of axiomatic systems, undefined terms are also called primitives. Although it
sounds like “double-speak,” these primitives are object names, but the objects they name
are left undefined. (The axioms are statements within the system that make assertions
about the primitives.) If a meaning is attached to a primitive, it is called an interpretation.
Undefined terms are also found in a mix of axiomatic systems and geometry in
which definitions are formed using known words or terms to describe a new word.
There are three words in geometry that are not formally defined—point, line, and
plane—because they can’t be described without using words that are themselves
undefined. These terms are fundamentally important in the study of geometry,
because they are needed to further describe even more complex objects such as circles
116 and triangles. (For more about geometry, see “Geometry and Trigonometry.”)
What are some well-known axiomatic systems?
O
ne of the most well-known axiomatic systems was developed by the Greek
mathematician Euclid (c. 325–c. 270 BCE). He presented 13 books of geome-
try and other mathematics titled Elements(or Stoicheionin Greek). Included in
these books were theorems about geometry and numbers derived from five pos-
tulates about points, lines, circles, and angles, four axioms about equality, and
one axiom stating “the whole is greater than the part.” A more modern axiomat-
ic system is the axiomatic set theory, which is based on eight axioms and three
undefined terms.