FOM: Communicating Minds 1

[Harvey Friedman]

BRIEF REPORT ON WORK IN PROGRESS. AN INFORMAL AND SEMIFORMAL DESCRIPTION OF
THE RESULTS WILL BE DISCUSSED LATER ON THE FOM.

We can think of our principal formal systems for f.o.m. in terms of a
single mind contemplating mathematical universes.

We can instead rework these principal formal systems for f.o.m. in terms of
two minds, each contemplating a mathematical universe, where the two minds'
universes do not coincide (but considerably overlap). In particular, we
have worked with the assumption that the second mind dominates the first
mind, so that the first mind's universe is a substructure of the second
mind's universe, but not vice versa.

This idea immediately and naturally leads to extensions of principal formal
systems for f.o.m. by various transparent ontological axioms and epistemic
axioms.

The resulting extensions are sharply stronger than the original systems, up
to mutual interpretability. For example, if we start with formal
arithmetic, then we get second order arithmetic (as a first order system)
and somewhat beyond. If we start with second order comprehension, then we
get ZFC and even ZFC with subtle cardinals, etcetera. If we start with
third order comprehension, then we get ZFC plus large cardinals based on
elementary embeddings. If we start with ZF, then we get ZFC + elementary
embedding from a rank into a rank, and even ZF + j:V into V.

The fundamental epistemic axiom asserts that the two minds agree on the
truth values of all statements that are about entities that are in common
to both minds. (In the cases being worked out, what is common to both minds
is exactly what is present in the first mind). Since the two minds'
entities do not coincide, it is natural to want this axiom of
communication.

We expect that this method of moving to two communicating minds should bear
fruit in many philosophical contexts where subjective notions are at issue.
In particular, in connection with vagueness. E.g., we expect some
fundamental principles about vagueness which reflect that different minds
have different extensions of vague concepts, but nevertheless can
effectively communicate about them - even a multitude of minds forming a
society which communicates effectively about vague concepts, with different
extensions, at the same time communicating with and about each other, and
about the society as a whole, as well as other societies. The plan is for
such systems to be mutually interpretable with ZFC, and with ZFC plus large
cardinals.

We can view the results obtained thus far as taking a subjective view - or
partly subjective view - of predication on objects, where the objects
themselves are not predicates. One could say that the notion of predicate
is being treated as vague. This suggests the possibility of a corresponding
general theory of vague concepts discussed in the previous paragraph.