FOM: Having truth-values
William Tait
wwtx at midway.uchicago.edu
Wed Oct 18 15:30:20 EDT 2000
Some of the discussion of CH and second-order logic has concerned the
question of whether this or that mathematical proposition A, e.g. CH,
has a truth-value, i.e., whether
(1) A is true or A is false [i.e. not-(A is true)]
is true. Here A is not a sentence of a formal system being
interpreted in a model: it is an ordinary meaningful mathematical
statement.
What meaning should the sentence ``A is true'' have, other than the
meaning expressed by A? The illusion that Steel mentioned (Oct 16) is
precisely the illusion that there is a further criterion of truth,
e.g. given by Tarski's definition of truth, where ordinary
mathematical language, of which A is a sentence, is thought of as
though it were a formal language (the object language) being
interpreted in some MODEL-IN-THE-SKY within some superlanguage (the
metalanguage). If one rejects this picture, as really we should, then
``A is true'' can have no sense other than that of ``A'' itself, in
which case (1) becomes
(2) A or not-A
which most of us accept as valid in mathematics.
So CH has a truth-value; but we are not in the position to determine
which one. We will be in the position to do so when and if new axioms
are accepted which imply CH or its negation.
On these grounds, I think that the assertion that CH, or its
truth-value is `indeterminate' can be nothing more than a
sociological prediction that such axioms will never be found. I don't
know how risky that prediction is.
Bill Tait
More information about the FOM
mailing list