高等微積分優筆記 1234.56.78

=Introduction=

Peano Axioms
The set of natural numbers, denoted by $$\mathbb{N}$$, satisfies:

There exists an element, called $$1 \in \mathbb{N}$$.For each $$n \in \mathbb{N}$$, there exists an unique element $$S(n) \in \mathbb{N}$$, called the successor of $$n$$, satisfying the following properties:For $$n, m \in \mathbb{N}, n \neq m$$, then $$S(n) \neq S(m)$$.$$1 \neq S(n) \; \forall n \in \mathbb{N}$$ (i.e. $$1$$ is not successor for any $$n \in \mathbb{N}$$).If $$A \subset \mathbb{N}$$ satisfies:$$1 \in A$$$$n \in A \Rightarrow S(n) \in A$$then $$A = \mathbb{N}$$.</li></ol></li></ol>