[FOM] Paradox on Ordinals and Human Mind

Harvey Friedman friedman at math.ohio-state.edu
Fri Dec 17 15:28:57 EST 2004

On 12/17/04 7:18 AM, "Dmytro Taranovsky" <dmytro at MIT.EDU> wrote:

> 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?


E.g., what is a human mind? Your mind, all presently living minds, or what?

Harvey Friedman

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