## Franklin's Notes

### Order type of the irrationals

The order type of the irrational numbers $(\mathbb R\backslash \mathbb Q,\leq)$ is denoted $\theta$. We have the following algebraic identities:

• $\theta^\ast = \theta$, as evidenced by the mapping $x\mapsto -x$ which preserves irrationality

• $\theta+\theta=\theta$, which can be shown by proving the intermediate result that the ordering of positive irrationals $(\mathbb R^+\backslash\mathbb Q,\leq)$ has order type $\theta$

• $\theta+1+\theta=\theta$, which can be shown by proving that $((x,\infty)\backslash\mathbb Q,\leq)$ has order type $\theta$ when $x$ is irrational

• $(1+\theta)\cdot\zeta=\theta$, which follows from the previous identity and the fact that $\theta\cdot\zeta=\theta$