Undefined Terms
By David A.
Smith
The first part of an axiomatic system is a list of undefined terms, which are the technical words that will be undefined and used in the subject.
Recall, Euclid attempted to define all his terms, but we now recognize that that is not an achievable goal.
For exampple, a standard dictionary appears to contain a definition of every word in a language, but there will inevitably be some circularity in the definitions.
So, rather than attempting to define every term, we simply state certain key words to be undefined and work from there.
In geometry, we usually take the words point and line to be undefined terms.
In another subject they might use words set and element of to be undefined.
Undefined Terms (Neutral Geometry) The following words
point, line, plane, space
are taken to be undefined terms; that is, we have a common agreement of their meaning which will not be explained nor or assumed to have any properties.
In any geometry which assumes the existence and properties of the real numbers (along with the basics of set theory) the axioms that govern these undefined terms should explicitly state how the relationships between these undefined terms and the real numbers work.
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