I am working my way through ordinal arithmetic and cardinality, so I wanted a quick reality check that I am correctly differentiating between structure preserving bijections (isomorphisms) and non-structure preserving bijections.
A consequence of this difference is the following statement:
$\omega \ncong \omega+1$ but $\omega \xrightarrow[\text{onto}]{1-1} \omega+1$, where $\omega$ refers to the set of all natural numbers
That is to say, there is no structure-preserving bijection between $\omega$ and $\omega+1$ (where the "structure" in question is $\in$ and its corresponding properties)...however $ \omega \approx \omega+1$ (i.e. there is a non-structure preserving bijection between $\omega$ and $\omega+1$).
Is this the right idea?
(There appears to be a similar post here: $\omega+1\sim\omega$ but $\omega+1\not\simeq\omega$ but I think the relationship $\sim$ defined by the author of the post presupposes that the individual is familiar with the Schroder-Bernstein theorem).
Correct. There can't be a structure-preserving isomorphism $\omega \to \omega + 1$. The latter has a maximal element $\omega$ but the former does not, so what $x$ could satisfy $f(x) = \omega$? Or rather, if we did have such an $x$, where would $x+1$ go?
But they are of the same cardinality, as you can easily see by taking the maximal element $\omega$ from $\omega + 1$ and shoving it at the beginning. That particular bijection doesn't preserve the order structure, but it is a bijection.