Unbounded Finity?

Question

Consider the successor of the largest finite ordinal that will ever be considered alone. But then it wasn't the largest finite ordinal that will ever be considered alone. How do we get around this paradox? The largest finite ordinal that will ever be considered alone does exist, and yet we can consider its successor.

Answer 1

This apparent paradox has nothing to do with ordinals, or metalanguage. It is a time travel/omniscience paradox. Let's strip away the unnecessary bits.

How many words will the longest sentence I personally will say tomorrow be? Maybe I predict seven, but that doesn't stop me from saying a sentence with eight words. No matter my prediction, I can break it.

You will note that the difficulty disappears if the time travel is taken away. If we ask about the largest ordinal that has previously been considered, that is a unique number (that periodically increases with time).

Answer 2

This is essentially Jules Richard's paradox. You define an ordinal in a language and then talk about its consequences in the metalanguage. But it's not problematic in the original language and doesn't have the desired property in the metalanguage.

Basically, the difficulty comes from the imprecision of natural language ("the largest finite ordinal that will ever be considered by itself") and its translation into a formal language where it can be unambiguously defined.