Fwd: [FOM] Question about hard-core independence

[William Tait]

By mistake, I sent this message privately to Haim Gaifman yesterday. It 
might be of general interest.

Begin forwarded message:

A small point:
On Friday, January 10, 2003, at 07:35  AM, Haim Gaifman wrote:

> Suppose that T is axiomatizable by a set of axioms that
> represent all that we accept as evidently true, under some
> presupposed interpretation of the language.

Perhaps its an issue that  can be avoided, but I would want to say that 
there is no such set of axioms: If I accept T, then I accept CONSIS_T, 
for example. On the other hand, there is a question about whether the 
absence of these sentences, whose acceptance is `implicit' in the 
acceptance of T---essentially reflection principles---will affect your 
point, namely

> A hard-core
> independence for T may raise some doubt whether our intended
> interpretation of the language really picks up a unique
> model (or a unique "world").

---at least when the independent sentence is the continuum hypothesis, 
since the addition of such new sentences may not affect independence. 
So far, in the case of CH, they haven't.

On the other hand, maybe there is no one theory such as you suppose 
because there are more than one, expressing different conceptions of 
set theory. For example, there might be, from one point of view, a 
plausible new axiom for which it is impossible that there be anything 
like a core model. (I have no idea if that is likely (?).) Then there 
would be a split between what Jensen (in some past issue of BSL) called 
the Pythagorian conception (which would reject the axiom) and the 
Newtonian conception of set theory.

Bill Tait