[catiopatio]

> 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.

[ironSkillet]

> They aren’t actually different sizes, though.

What's your precise definition of size that allows someone to actually make a rigorous argument comparing the size of any two sets?

[catiopatio]

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.

[ironSkillet]

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.

[catiopatio]

> So the whole debate is moot.

Well, yes. :)

[ironSkillet]

Thanks for wasting my time by avoiding any precision in your language lol.