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> 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 t
by catiopatio 3y ago
> 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.
- deleted 3y ago[deleted]
- ironSkillet 3y ago> 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 3y agoI 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 3y agoIf 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 3y ago> So the whole debate is moot. Well, yes. :)
- ironSkillet 3y agoThanks for wasting my time by avoiding any precision in your language lol.