In the integers:1,2,3... we can say that there isn't a largest number; but it turns out with the advent of set theory we can posit a largest number omega that doesn't lie within the integers and yet bounds them. In mathematical terms we can say that it is a limit to them.
Of course, with the advent of Cantorian set theory, it was also realised that one can define and investigate larger infinite numbers, which are ordinals and cardinals (cardinals are certain distinguished points in the ordinals and are very sparse). Then one can ask the same question at his new level: Is there a largest Cardinal?
Now, it turns out that Reinhardt cardinals are the largest cardinals yet defined in ZFC; To refine the question - are they provably the largest possible? Is there in fact provably, or conjecturally a largest cardinal wrt ZFC? (My intuition would say that there isn't).
(Choice is generally useful when dealing with properly infinitary objects, which is why I've asked with reference to ZFC, rather than ZF);
NBG, and NFU are alternative set theories that make formal use of a concept that ZFC informally uses, that is classes; as classes are not sets then formally the notion of cardinality or ordinality is applicable to them - but one could argue with judicious changes in the axioms that one could extend the notions to here; it would then seem a possibility of defining a class-cardinal that is the limit of all set-cardinals.
Is this this actually possible?
But then, one could argue that the same process could be iterated; in a much more profound sense the notion of mathematical infinity is always in potentia, though there are points at which we can say a certain actuality has been achieved (the limit points).
Is this a good argument?