Does the set ω + 1 represent a higher infinity that ω? In other words, is ω + 1 countable? What about ω +ω?

Question

I answered the first part as follows:

$\omega + 1 = \{0, 1, 2, 3, 4, 5, ..., \omega\} = \aleph_0 $ Since every element of ω is finite and one element in $\omega + 1$
is infinite, they are clearly different infinite numbers. This is also known as an ordinal number. Hence, $\omega +1$ is countable but it doesn't represent a higher infinity. It is infinity itself.

But when it comes to $\omega + \omega$, I'm a bit confused on how to approach it.

Answer 1

$\omega+1$ is an order type (it represents a countable well-order with a unique maximum beyond a copy of $\omega$ while $\omega+\omega$ is two copies of $\omega$ ordered one left to the other. Both represent countable sets in cardinality, so they're not a higher order of infinity.

The set of all countable well-orders is of a higher cardinality, called $\omega_1$ (as a well-order) and $\aleph_1$ (as a cardinal number).