FOM: realism and bivalence
neilt at mercutio.cohums.ohio-state.edu
Fri Jan 30 16:11:41 EST 1998
John Steel answers Harvey Friedman as follows:
>Realism asserts that there are sets, and hence ... "there are sets" is true.
> ... Whether this is Godelian naive realism I don't know.
>I would subscribe to "either there is a measurable cardinal or there is
>no measurable cardinal".
Comment 1. One needs to distinguish ontological realism from semantic realism.
The first of Steel's claims just quoted expresses a commitment only to
ontological realism. Semantic realism would go further. According to the
semantic realist, every sentence is either true or false of "the world",
i.e. (in this case) the intended model of set theory, independently of our
means of coming to know what its truth-value is. Saying that there are sets
is not yet enough to get you determinate truth-values for all statements
Comment 2. Being prepared to assert every instance of LEM (such as "either
there is a measurable cardinal or there is no measurable cardinal") is usually
justified by appeal to the following position: in every model M of my axioms,
and for every sentence S, either S is true in M or S is false in M. This is
the principle of bivalence, the litmus thesis of semantic realism. Such
semantic realism does not yet amount, however, to the *naive* realism of,
Comment 3. The naive realist thinks in addition that there is some unique,
preferred, intended model of those axioms. Following the usual practice, let
us call this intended model V. Naive realism makes the status of the existence
of, say, a measurable cardinal much more pointed: IS THERE A MEASURABLE
CARDINAL ***IN V***? For the naive realist, there is a correct yes/no answer to any such question about V.
Philosophically, we have:
ontological realism < semantic realism < naive realism
Of course, some peculiar hybrid positions are logically possible. The naive
realist might think there are correct yes/no answers to all questions about V,
while not holding that the same is true of models *other than* V! But I do not
know of anyone who puts forward this bizarre combination of metaphysical views.
The foregoing crescendo ignores these bizarre positions.
More information about the FOM