[FOM] Ultrafinitist notion of "open problem"

Nik Weaver nweaver at math.wustl.edu
Wed May 27 12:56:37 EDT 2015

Tim Chow asked:

> But does this necessarily mean that
> Nelson regards "PA is consistent" as directly meaningful?

A few years ago Ed Nelson gave a talk at a MidWest PhilMath Workshop
about his attempts to show that PA is inconsistent, and during the
question period I asked him whether this meant he agreed that the
statement "PA is consistent" has a definite truth value.  I didn't
fully understand his answer, so the most reliable thing I can say is
that he didn't give a clear "yes" answer to this question.

As I remember it, his response was something to the effect that
the statement "PA is consistent" has some kind of pragmatic meaning,
in that a specific inconsistency might one day be found, but that
does not imply that it is "meaningful" in the sense of having a
definite truth value.  But as I said, I didn't really understand
him, so take that interpretation with a grain of salt.

> If we accept this gloss, then the next question I have is, is the
> consistency of PA an open problem of the type that can be settled only in
> one direction?

And based on my interpretation of this incident, I suspect the answer
to this question would be "yes".

Nik Weaver

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