[FOM] Paradox on Ordinals and Human Mind

[Dmytro Taranovsky]

What is the least ordinal that cannot be identified by a human mind?

Some human thoughts refer to ordinals, while others do not.  Since for
every non-empty predicate P on ordinals, there is the least ordinal
satisfying P, one can meaningfully ask about the least ordinal whose
definition or identification is beyond the potential capabilities of
minds.  However, this description appears to identify the ordinal, and
hence contradict itself.

Note that because the description refers to possible capabilities as
opposed to current reality, one cannot escape by claiming that the
ordinal is time dependent or that it depends on future contingencies.

There are three ways to address the paradox:
1.  Infinite sets do not exist, but humans can define arbitrarily large
integers.
Or
2.  Word "identify" and certain other words are meaningless (at least in
the sense they are used in the paradox). 
Or
3.    The potential of the human mind extends beyond the finite, and
every ordinal can be identified by a human mind.

Which resolution is correct?

Dmytro Taranovsky