User blog comment:Alemagno12/Unthinkable numbers/@comment-24920136-20170418023237/@comment-30754445-20170418172000

To be more formal:

The mathematical object he is thinking about is the set of these numbers (assuming for the moment that we have a formal definition for "unthinkable number").

He is not thinking of specific members of this set, so there's no paradox. The set itself is thinkable. The numbers in it are not.

This reminds me of the fact that there are real numbers which have no finite description at all (this follows directly from the fact that the c.> ℵ₀). Yet the definition of the complete set of real numbers is relatively short and simple. Once again the set as a whole is a much "simpler" complex then the vast majority of its members.