Is $\omega + \omega^2 = \omega^2$ true?

Question

Just a simple question. Is $\omega + \omega^2$ equal to $\omega^2$, I’ve just been thinking about it and if that’s false and it equals $\omega^2+\omega$ then we could define a set of countable ordinals using Cantor’s diagonal argument with cardinality $2^{\aleph_0}$ which would prove the continuum hypothesis, I think, which means that because the continuum hypothesis is unsolvable under ZFC that $\omega + \omega^2$ must just equal $\omega^2$. Unless I’ve done something wrong (which is likely), which is why I’m asking the question.

Answer 1

Addition of ordinals is non-commutative. Additive elements can “combine together” to equal the largest in the sum. In general, elements combine when the one on the left is smaller than the one on the right. So $\omega^2 = \omega + \omega^2 = \omega^2\neq \omega^2 + \omega$. This is covered on wikipedia with diagrams showing why it comes out this way.

Note that at least one element has to be infinite for it to combine. It’s still the case that $3+5=8=5+3$. Finite and infinite numbers combine, so $3+\omega=\omega\neq\omega+3$