[FOM] Foreman's preface to HST

[Roger Bishop Jones]

On Tuesday 27 Apr 2010 08:16, joeshipman at aol.com wrote:

> This is merely a practical observation; if you know of
>  any serious mathematical investigation of questions
>  which can be stated in the language of set theory that
>  cannot easily be conducted in the framework of
>  first-order-logic + ZFC, I'd like to hear about it.

There are conceptual advantages which flow from construing 
ZFC (or other first order axiomatisations of set theory) in 
second or higher order logic.  These are conceptual 
advantages which have practical significance in making 
problems definite which otherwise are not.

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.

Discussions of CH are too often conducted in a context in 
which not only is the deductive system incapable of settling 
it, but the semantics also fails to de definite on its truth 
value.

Alternatively you may construe this point as the observation 
that strictly speaking some of the important unsolved 
problems can not "be stated in the language of set theory", 
as it is normally understood.  There still are such cases in 
second order set theory, but it is still a considerable 
advance, semantically.

Roger Jones