[FOM] Fact and opinion in F.O.M.

[Timothy Y. Chow]

Joe Shipman wrote:

> (1) all the arithmetical consequences of all the axioms that have ever 
> been proposed appear to be compatible with each other.
> (2) this is completely untrue for statements of higher type.

Okay, how about the following suggested definition of "fact"?

   (*) A mathematical statement X is a fact if no axiom that has ever
       been seriously proposed (or ever will be seriously proposed)
       implies not-X.

This stays closer to your statement (1) above and steers clear of my 
objections to your "permanent disagreement" formulation.

If we accept, as I do, that V = L has been "seriously proposed" as an 
axiom, then ~CH is not a fact.  Off the top of my head, I can't think of a 
seriously proposed axiom that implies ~CH, but maybe someone else can; if 
there is one, then that would mean that CH is not a fact either. 
(Freiling's axiom of symmetry, maybe?)  If neither X nor not-X is a fact 
then we could say that X is not a "matter of fact" (and similarly not-X is 
not a matter of fact).

Then your question becomes whether there exist any non-arithmetical facts.

By the way, there seems to be some similarity between your concept of 
"fact" and Feferman's concept of a "definite mathematical problem," as in 
his paper, "Is the continuum hypothesis a definite mathematical problem?" 
(Though Feferman seems to take a different direction from what you're 
proposing.)

Tim