FOM: CH and 2nd-order validity

[Roger Bishop Jones]

Many thanks to those who responded to
my question about CH and 2nd-order validity.

Robert Black said:

>I'm not quite sure what this means, since after 'independent of'
>I expect the name of a statement or set of axioms...

and some other respondents made similar complaints.

The "set of axioms" I intended was the axioms of ZF + the translations into
the
language of ZF of the valid sentences of 2nd-order logic.
(glossing over possible differences in the semantics of the power set
constructor in the two languages; an effect of the extra axioms being to
force
the "standard" interpretation of the power set constructor in ZF(maybe!))

I hope this does not change the consensus view, which seemed to be
that it is established that CH is not independent of 2nd-order validity.

The question was provoked by my beginning to read Stewart Shapiro's
most recent book "Thinking about Mathematics", in which he mentions
that some philosophers regard the truth value of CH as not determinate.
(which shows that sentences which convey no news may yet
provoke action)

If the language ZF is construed as being about all the models of ZF,
then CH does not, indeed, have a determinate truth value.
However, at least two respondents to my question asserted or implied
that CH does have a definite truth value, and reading between their
lines I guess that they were admitting only standard interpretations of ZF
in which power sets are complete.

It seems to me that "the problem" of CH may be considered in
three parts.

1.  First it is desirable to make the semantics of first order set theory
sufficiently precise that CH has a definite truth value under that
semantics.

2.  Then there is the problem of deciding what that truth value is.

3.  Finally one could consider candidate axioms to add to
ZF to enable CH or its negation to be derived (as appropriate).
(CH or its negation would do!)

Of course, the only (and very) difficult step is (2),
and I have nothing to offer here.
Doing (1) first and explicitly is mainly desirable to avoid spurious
philosophical disputes.

Reflecting a little longer on (1), it seems that settling on a notion of
standard model for ZF similar in spirit to the idea of standard model
in second order logic suffices to make the truth value of CH determinate
(though not to make clear what it is).
This I take to be equivalent to adding the second order validities as
axioms.

Since 2nd-order validity is regarded in some quarters with suspicion
it might be worth asking whether there are any weaker and less
controversial semantic notions which suffice to settle CH.

For example:
    Is CH independent of true arithmetic?

(and if this were not the case, would it suffice to interpret
set theory in well-founded model's of ZF for CH to be determinate?)

Roger Jones
RBJones at RBJones.com