What does $\alpha+\gamma$ mean when $\alpha$ and $\gamma$ are well-ordered sets?

Question

I was asked to prove the following:

let $\gamma$ be a well ordered set with the following property: for any $\alpha$ and $\beta$ well ordered sets, if $\alpha+\gamma=\beta+\gamma$ then $\alpha=\beta$. Show that $\gamma$ is a finite set.

This might be a very doable question to solve, but I don't understand it. What does $\alpha+\gamma$ mean? they are not numbers, they are sets. What does this notation mean?

Answer 1

Ordinals numbers are numbers. They have their addition and multiplication and even exponentiation.

The notation $\alpha+\gamma$ means that we take the ordinal $\alpha$ then we concatenate it with $\gamma$. Formally speaking, this is the only ordinal $\eta$ which is isomorphic to the lexicographic order on: $(\{0\}\times\alpha)\cup(\{1\}\times\gamma)$.

One can also define addition by induction. Let $s(\alpha)$ denote the successor ordinal of $\alpha$, then

1. $\alpha+0=\alpha$,
2. $\alpha+s(\beta)=s(\alpha+\beta)$,
3. $\alpha+\delta=\sup\{\alpha+\gamma\mid\gamma<\delta\}$ when $\delta$ is a limit ordinal.

Answer 2

Given well-ordered sets $\alpha$ and $\gamma,$ the set $\alpha+\gamma$ is the disjoint union of $\alpha$ and $\gamma,$ ordered with all the elements of $\alpha$ first (in their given order), followed by all the elements of $\gamma$ (in their given order). This ordering is in fact a well-ordering of the disjoint union.

As a hint for how to prove it, suppose $\gamma$ is infinite, and consider the case that $\alpha$ is empty and $\beta$ is a singleton. You should be able to show that $\alpha+\gamma$ and $\beta+\gamma$ are order-isomorphic (so identical, in this context).