User blog comment:Alemagno12/Unthinkable numbers/@comment-24920136-20170418023237/@comment-30754445-20170418175702

True (for integers and ordinals).

The usual work-around, as Simply Beautiful Art said, is to speak of definability in a specific system.

With "unthinkable numbers" we have the problem that the full details of the system itself (the human mind) is unknown. What's worse, different humans have different "specifications" so there isn't a single well-defined system we can refer to.

So yeah, the whole concept is pretty fuzzy in practice. But this doesn't change the basic conclusion that if a human brain has EE16 states then a human brain can only represent EE16 different numbers. We have no practical way to tell which numbers they are (a feat which - by its very definition - is beyond the human mind to accomplish) but we can still reach logical conclusions about the set of these numbers (among other things, we can use tge pigeonhole principle to deduce that there are numbers in the neighborhood of EE16 which have cannot be represented in our mind at all).

By the way, the whole "smallest undefinable number" trick can only work for stuff which is well-ordered. So it works for integers and for ordinals, but it doesn't work for the reals. There are real numbers which are completely undefinable, and none of these numbers is "the smallest".