Axioms

• Weak Mathematical Induction: If the following two things can be shown about a proposition $P(n)$:
  • Base Case: $P(1)$ is true.
  • Induction Step: If $P(n)$ is true, then $P(n+1)$ is true.
  then $P(n)$ is true $\forall n\in{\bf N}$.
• Strong Mathematical Induction: If the following two things can be shown about a proposition $P(n)$:
  • Base Case: $P(1)$ is true.
  • Induction Step: If $P(k)$ is true $\forall k\in{\bf N}$, $1\leq k<n$, then $P(n+1)$ is true.
  then $P(n)$ is true $\forall n\in{\bf N}$.
• Pigeonhole Principle: If there are $n+1$ pigeons and they are put in $n$ pigeonholes, then there is a pigeonhole that contains at least two pigeons.
• Well Ordering Principle (WOP): If $S$ is a non-empty set that is a subset of the positive integers, then $S$ has a least element.
• Transitivity: If $a=b$ and $b=c$, then $a=c$.
• Substitution: If $a=b$, then $a$ can be substituted for $b$ in any statement without changing the veracity(truthfulness) of the statement.
• Distributive Property: $ab+ac=a(b+c)$.