\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \newcommand{\E} {\mathrm{E}} \)
deepdream of
          a sidewalk
Show Question
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \)
Math and science::Analysis::Tao::02. The natual numbers

Tao's 5 axioms for natural numbers

1. 0 is a natural number.
The root node.
2. If n is a natural number, then n++ is also a natural number.
The increment operation. The only axiomatic operation.
3. If \( n++ = m++ \) for any natural numbers n and m, then \( n = m \). (Different natural numbers must have different successors).
Prevent loops where there are two nodes that increment to the same node.
4. \( n++ \ne 0 \) for every natural number n. (0 is not the successor of any natural number).
Prevent a circle loop back to 0 (not covered by 3). 
5. Let P(n) be any property pertaining to a natural number. Suppose that P(0) is true, and suppose that whenever P(n) is true P(n++) is also true. Then P(n) is true for every natural number n.
There are no other connected components other than the one containing 0. A stange approach to address this. Tao mentions that this is more of an 'axiom schema' in logic parlance.