[FOM] Paradox on Ordinals and Human Mind

Schneelocke schneelocke at gmail.com
Fri Dec 17 14:32:54 EST 2004

> 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?

I'd say the second one is - the concept of what "cannot be identified
by the human mind" is not well-defined in a mathematical sense. I
think I recall reading some thoughts on this by Raymond Smullyan,


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