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

[Noah Schweber]

> Off the top of my head, I can't think of a
seriously proposed axiom that implies ~CH

The inner model hypothesis and forcing axioms above MA each imply the
failure of CH. I don't know what the bar is for "seriously proposed," but
my own instinct is that each of these meet it (although I'm very generous).

On Thu, Dec 26, 2019 at 2:45 PM Timothy Y. Chow <tchow at math.princeton.edu>
wrote:

> 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
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