> It is in this sense that there are infinities of different sizes.
They aren’t actually different sizes, though.
All this proves is that under specific set theoretic assumptions, a contradiction arises if you define “size” as “cardinality” and assume that a particular bijective relation exists between your two infinite sets.
It doesn’t actually mean the sets have different sizes, it just means they differ under a set of assumptions that may (or may not) be useful for your purposes.
I don’t have one that doesn’t admit a contradiction here … which I’d argue is because comparing the size of an infinite set is nonsensical, even if the properties used to do so are otherwise useful.
Similarly, I can also work around Russel’s paradox by introducing infinite universes, but that doesn’t actually resolve the paradox, it just provides a set (ha ha) of rules that may be leveraged to formalize the Set category and otherwise prove useful things.
Just because your formalization admits a proof by contradiction doesn’t actually prove two infinite sets have different sizes, it just proves that a contradiction exists under your assumptions.
If you aren't allowed to operate in a logical system with a concrete definition of "size", then you can't say things like "doesn't actually prove two infinite sets have different sizes". So the whole debate is moot.
They aren’t actually different sizes, though.
All this proves is that under specific set theoretic assumptions, a contradiction arises if you define “size” as “cardinality” and assume that a particular bijective relation exists between your two infinite sets.
It doesn’t actually mean the sets have different sizes, it just means they differ under a set of assumptions that may (or may not) be useful for your purposes.