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

[Joe Shipman]

The strong Fubini axioms I have proposed, which generalize Freiling, and the real valued measurable cardinal axiom, which implies them, as well as other “Product measure extension axioms” that I have seen, all contradict CH. I am sure there are others that have been proposed unrelated to the Measure problem which also contradict CH.

Your proposal seems more of a property that mathematical facts must have, than an applicable definition.

At the moment, there aren’t any propositions other than arithmetical consequences of ZF that I would be willing to assert are “mathematical facts”, but I’d be happy to have someone argue either that some such are not facts, or that some non-arithmetical statements are matters of fact. 

If a non-arithmetical statement is both a matter of fact, and provably equivalent over a weak enough theory to an arithmetical consequence of ZF, I would be willing to call it a fact, but I’m not sure how weak is weak enough, and I’m not sure whether there are any such “matters of fact”. For example, one could claim that there is no such thing as an uncountable set, or no such thing as an infinite set, and not be wrong about anything verifiable by any of the tools I am currently familiar with. 

If fundamental physics were in better mathematical shape, the situation might be different and a Putnam-type argument could establish infinities in an ontologically convincing way, but it isn’t there yet.

— JS

Sent from my iPhone

> On Dec 26, 2019, at 5:50 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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