>>16709267
>like with infinity?
Yes that’s pretty much it. An inaccessible cardinal is an ordinal that you can’t reach by applying basic operations to smaller ordinals, just like you can’t reach [math]\omega[/math] from by applying basic operations to ordinals [math]\alpha_1,\ldots,\alpha_n<\omega[/math] aka natural numbers.
> we just saying they exist and working from there? like an axiom declaring "suppose this cardinal exists"?
Yeah exactly. If ZFC is consistent, then it has models in which there is no inaccessible cardinal, and other models in which there are inaccessible cardinals. So when we want one we just add an extra axiom (or really, an axiom scheme).
That’s how it works pure-mathematically when you don’t worry about philosophical hangups like whether these ordinals “do” exist or “should” exist. Personally I think infinity is fake and gay, but it definitely is a useful abstraction for talking about really large and small numbers. I hope something like that happens with these crazy big cardinals too, but who knows, I’m not sitting through some crazy Shelahshit or Woodinshit for 1000pp just to find out the answer is “wait, you wanted this to be even a little bit useful? why??”
