FOM: CH and 2nd-order validity

John Steel steel at
Tue Oct 17 15:03:42 EDT 2000

On Tue, 17 Oct 2000, Robert Black wrote:

> 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.

     I realize this is beside your point, but God and the visibility of
set have nothing to do with Platonism.

      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).

      One thing some set theorists say is "V is not equal to L". (They are
actually more interested in stronger, more useful assertions which imply
V not equal L.) This is false if we interpret "set" to mean the same as
"set in L", and epsilon to stand for membership.

      The Skolem's paradox arguments for some indeterminacy of meaning are
based on there being elementarily equivalent (satisfying all the same
sentences) but nonisomorphic models of any T having an infintite model, so
they don't seem directly relevant here. 

> 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'?
          First, I mildly regret the word "baloney", since if I write
long enough on the subject I'll no doubt produce my share.
          The basic line of the Skolemites, that one cannot assert
that there is an uncountable set, is best refuted in the style of
Dr. Johnson's kicking the stone. In fact, we assert there are
uncountable sets all the time. Your theory of meaning is just wrong if it
implies that no one can assert that there are uncountable sets, or sets
not in L. 

         I haven't read Putnam's "Models and Reality" in a long time
(was that the paper you were referring to?), but as I remember it
basically just fell into some Skolemite error.

John Steel

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