FOM: Re: CH and 2nd-order validity

[Roger Bishop Jones]

In response to: Harvey Friedman Sunday, October 15, 2000 8:55 PM

> First of all, I want to say that this whole thread concerning CH and
second
> order validity consists of observations and comments that have been well
> known for38 (Cohen) to 62 (Godel) years.

In mitigation I point out that almost all the messages were responses to
my initial enquiry which began "Is it known whether ....".

Thanks to Harvey for answering some of the questions I raised in my
second message on this topic.
However, in relation to the central concern of my message, the desirability
of being more explicit about the semantics of first order set theory,
Harvey's message compounded my confusion, so I am here seeking clarification
of his position.

Responding to my::

>>If the language ZF is construed as being about all the models of ZF,
>>then CH does not, indeed, have a determinate truth value.

Harvey says:

> ZF is not about models of ZF. ZF is about sets.

and later:

> The first order semantics of first order set theory are no different than
> the first order semantics of predicate calculus generally. But this does
> not have anything to do with definite truth values of statements such as
CH.

I find it difficult to reconcile these two statements.

Does CH have a truth value under the semantics of first order set theory?
If that semantics is exclusively determined by the semantics of first order
logic and the axioms of first order set theory (as Hilbert might have
insisted) then it does not.

I expect the semantics of first order set theory to give the truth
conditions for all sentences of first order set theory, including CH.
How can the semantics of first order set theory not have anything to do with
the truth value of CH?

Further compounding my difficulty in understanding Harvey on this matter is
the fact that he frequently talks of the semantic consequences of 2nd-order
ZFC.
The fact that he mentions "2nd-order" here suggest to me that he considers
these to be different from the semantic consequences of 1st order ZFC.
This makes perfect sense to me if first order ZFC is construed as being
about the first order models of the axioms of ZFC, but since Harvey has
ruled this out I am now at a loss to understand the difference between the
first order and second order semantics of ZFC.

It appears that neither true arithmetic nor well-foundedness settles CH.
Am I right to believe that true arithmetic is strictly weaker than
well-foundedness?

> CONJECTURE. Any reasonable weakening of 2nd order semantics results in a
> formulation of ZFC which does not decide CH.

Without some indication of what is meant by "reasonable" not much can be
made of this, however, since CH is much weaker than 2nd-order validity there
clearly are (possibly unreasonable) weaker semantic notions than 2nd-order
validity which suffice to decide CH (a trivial example being CH itself).

Roger Jones
RBJones at RBJones.com