[FOM] Foreman's preface to HST

Roger Bishop Jones rbj at rbjones.com
Fri Apr 30 01:16:27 EDT 2010

On Thursday 29 Apr 2010 05:19, Monroe Eskew wrote:
> On Tue, Apr 27, 2010 at 10:50 PM, Roger Bishop Jones 
<rbj at rbjones.com> wrote:
> > The advantage is that the semantics is more definite
> > and many questions which are independent of ZFC, for
> > example CH, can be seen to be settled (though we don't
> > know which way) by the standard semantics of second
> > order logic.
> I see this as a disadvantage.  There's no way of knowing
>  whether CH is true in second order set theory, yet
>  there's no way of investigating what the universe would
>  look like if it were true or false by producing
>  different models.  First order logic allows you greater
>  ability to investigate the logical and structural
>  relations between things.  If you want CH to be
>  "settled" and given a definite truth value, then you can
>  just say it's settled by a sufficiently *correct*
>  interpretation of first order ZFC, i.e. an epsilon
>  structure that includes *all* the subsets up to a
>  certain rank.

And how does that differ from doing second order set theory?

These are what I would call the "standard"  models of set 
theory, and the only reason why I talk about second order 
set theory is that the term "standard" is used in this 
context with that sense, whereas the term "standard" appears 
to be used in other ways when talking about interpretations 
of set theory.

Roger Jones

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