FOM: CH and 2nd-order validity

[Robert Black]

John Steel is surely right in saying that formal semantics can only be used
to model one language in another and that ultimately the meaning of the
language of set theory is given by its use. The problem is that it's
unclear whether or not that use gives determinate truth values to sentences
undecided by the axioms of set theory. For example, there is the following
argument (paraphrasing Hilary Putnam in JSL 1980):

Suppose for reductio purposes that Platonism is true and that in the real
universe of all sets visible from a God's-eye-view, CH (and hence V=L) is
false. And suppose God looks down on the use of set-theoretic language to
be found among practitioners of ZFC. He notices that everything they say
makes perfect sense if their word 'set' is taken to refer to all the
inhabitants of V, but would also make perfectly good sense were it taken
just to refer to the inhabitants of L. Since nothing determines one
interpretation as being right and the other wrong, and since under one
interpretation CH is false and on the other it is true, 'CH' as spoken by
the human set theorist has no determinate truth-value (and thus our
original, human-expressed, supposition has no clear meaning).

I don't myself accept this argument, but I do take it seriously. Would it
be an example of what Steel refers to as the 'baloney written on Skolem's
paradox'?

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845