Full Idea
First-order logic is hopeless for discriminating between one infinite cardinal and another.
Gist of Idea
First-order logic can't discriminate between one infinite cardinal and another
Source
Wilfrid Hodges (Model Theory [2005], 4)
Book Reference
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.11
A Reaction
This seems rather significant, since mathematics largely relies on first-order logic for its metatheory. Personally I'm tempted to Ockham's Razor out all these super-infinities, but mathematicians seem to make use of them.