[FOM] Foreman's preface to HST

[Monroe Eskew]

On Thu, Apr 29, 2010 at 10:16 PM, Roger Bishop Jones <rbj at rbjones.com> wrote:
>
> 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

The difference is that you don't restrict your entire theory of
mathematics from the outset and prevent yourself from accessing all of
the powerful tools of first order logic and set theory:  Completeness
and compactness theorems, Lowenheim-Skolem theorems, forcing, taking
inner models, taking ultrapowers, etc.  If you say the main advantage
of second order set theory is its semantic definiteness and thus
"settling" some questions, then I'm saying you can still view those as
"settled" in the same sense by simply saying they are settled by what
a "standard model" says of them.  (Or if you are bold enough, or
realist enough, just whether they are true.)  And then you proceed
with first order set theory and all of its great methods and results.

Monroe