[FOM] The Field with One Element?

[Mario Carneiro]

On Fri, Jun 17, 2016 at 2:41 PM, Andrius Kulikauskas <ms at ms.lt> wrote:

> If there was an alternate foundations of mathematics which yielded a
> helpful, meaningful, fruitful interpretation of F1, would that count in its
> favor?  And if it could do everything that FOM can do, then might it be
> preferable, at least for some?  But especially if that interpretation was
> shown not to make sense in other FOMs?
>

It is my belief that the discovery of an alternative FOM in which F1 makes
sense will be readily translatable into ZFC / "standard FOM", thus yielding
a "true" theory of F1. There is no chance of this alternative being
incompatible with ZFC in all senses - at the very least one can simulate
the formulas of the alternative FOM in ZFC and hence get a model for
whatever properties F1 is discovered to satisfy. In practice the embedding
will probably be much more faithful, and will probably boil down to
something recognizable to algebraists: a set with some relations and
operations.

A comparison with the discovery of ZFC models of HoTT seems relevant here.

Mario Carneiro
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