Foundations of Mathematics

Axiomatic System

What is an undefined term?

In terms of axiomatic systems, undefined terms are also called primitives. And 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 cannot 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 and triangles. (For more about geometry, see “Geometry and Trigonometry.”)


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