# Category:Definitions/Mathematical Induction

$(1): \quad$ A statement $Q$ is established as being true for some distinguished element $w_0$ of a well-ordered set $W$.
$(2): \quad$ A proof is generated demonstrating that if $Q$ is true for an arbitrary element $w_p$ of $W$, then it is also true for its immediate successor $w_{p^+}$.
The conclusion is drawn that $Q$ is true for all elements of $W$ which are successors of $w_0$.