In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or other formal system, the role of a primitive notion is analogous to that of axiom. In axiomatic theories, the primitive notions are sometimes said to be "defined" by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress.
Examples. In:
- Naive set theory, the empty set is a primitive notion. (To assert that it exists would be an implicit axiom.)
- Peano arithmetic, the successor function and the number zero are primitive notions.
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