Peano axioms Giuseppe Peano Mathematical induction Structural induction Recursive definition Naive set theory Element (mathematics) Ur-element Singleton (mathematics) Simple theorems in the algebra of sets Algebra of sets Power set Empty set Non-empty set Empty function Universe (mathematics) Axiomatization Axiomatic system Axiom schema Axiomatic method Formal system Mathematical proof Direct proof Reductio ad absurdum Proof by exhaustion Constructive proof Nonconstructive proof Tautology Consistency proof Arithmetization of analysis Foundations of mathematics Formal language Principia Mathematica Hilbert's program Impredicative Definable real number Algebraic logic Boolean algebra (logic) Dialectica space categorical logic