[FOM] Foreman's preface to HST

[Monroe Eskew]

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.