[FOM] Foreman's preface to HST

[Monroe Eskew]

On Sat, May 1, 2010 at 11:13 PM, Roger Bishop Jones <rbj at rbjones.com> wrote:
> My only reason for mentioning second order logic in this
> context was (I am repeating myself here, tying to be more
> explicit), that the phrase "standard model of ZFC" does not
> have a definite accepted meaning which corresponds to "model
> of second order ZFC" and the latter phrase is therefore
> less likely to be misunderstood.
>
> I would be delighted if there were a general recognition
> that the idea of a "standard model" of ZFC is important and
> that unsolved questions such as CH should be interpreted in
> that context rather than supposed to be lacking a truth
> value, (or worse, supposed to have a truth value without any
> semantic clarification).
>

1) Here's a definition: A "standard model" is any rank V_\alpha which
satisfies ZFC.  For instance if \kappa is inaccessible then V_\kappa
satisfies it.  The reflection theorem implies that if there is an
inaccessible kappa then in fact there is an \alpha < \kappa such that
V_\alpha is a standard model, and the least one has countable
cofinality.  (A Skolem-like paradox?)

2) Within the theory of ZFC (or NGB, MK, etc), CH is a well defined
statement that is either true or false.  Its quantifiers range over
the class of all sets.  However it is equivalent to make the
quantifiers range over V_{\omega+4}.

Monroe