Finite ordinal Exponentiation

Question

I confused a little when i do arithmetic on ordinals especially multiplication is what i wrote right? :

$(ω+1)$ = {$0,1,....ω$}

$(ω+1)(ω+1)$ = sup({lexicographic Order($(ω+1)×(ω+1)$)}) = $ω²+1$

$(ω+1)(ω+1)(ω+1)$ = sup({lexicographic Order($(ω+1)×(ω+1)×(ω+1)$)})= $ω³+1$

$(ω+1)ⁿ$ = $ωⁿ+1$

and

$(ω+k)ⁿ$ = $ωⁿ+k$

Im sorry for asking it because i couldn't find any library for python or website to check my calculation is right or not.

Thanks.

Answer 1

Ordinal multiplication has the property that $\alpha\cdot(\beta+1)=(\alpha\cdot\beta)+\alpha$ for all ordinals $\alpha$ and $\beta$, so

$$(\omega+1)\cdot(\omega+1)=\big((\omega+1)\cdot\omega\big)+(\omega+1)\;;$$

and $(\omega+1)\cdot\omega=\omega^2$, so $(\omega+1)\cdot(\omega+1)=\omega^2+\omega+1\ne\omega^2+1$.

Then

$$\begin{align*} (\omega+1)^3&=(\omega+1)^2\cdot(\omega+1)\\ &=(\omega^2+\omega+1)\cdot(\omega+1)\\ &=(\omega^2+\omega+1)\cdot\omega+\omega^2+\omega+1)\\ &=\omega^3+\omega^2+\omega+1\;. \end{align*}$$

Can you correct the rest of it from here?