@whiskyblanket,

Tautology:

*a = a*

Theorem:

*a + b = b + a *(

*= c*)

Thinking of Godel, the tautology is the inherently unprovable statement that allows completeness, that is, it's an axiom. To Kant (if I'm reading the

*Prolegomena* right), a theorem can be presented in two ways:

*a + b = b + a* (analysis or restatement of subject in predicate) or

*a + b = c* (synthesis or the predicate being a wholly different concept derived from the subject). For example, the statement

*A line is the shortest possible distance connecting two points* is a tautology (Euclidean axiom) whereas the statement

*A line is a sequence of points extended in one direction* is a theorem based on said tautology (note lines can be either finite or infinite).