[FOM] irrational conjectures

[Harvey Friedman]

On Mar 11, 2009, at 10:52 AM, Timothy Y. Chow wrote:

> Harvey Friedman wrote:
>> WILD CONJECTURE. There exists a positive integer n < 2^1000 such that
>> the statement sin(2^[n]) > 0 can be proved using (commonly studied)
>> large cardinals using at most 2^20 symbols, but cannot be proved in  
>> ZFC
>> using at most 2^2^2^20 symbols.
>
> Obviously, one can take any unsolved problem and formulate a similar
> wild conjecture.  But is there any particular reason to believe that
> large cardinals are lurking here?

Note that sin(2^[n]) > 0 can be proved in extremely weak fragments of  
arithmetic, no matter what n is.

Saying that "there exists n < 2^1000" radically distinguishes it from  
just routinely taking any open statement A, and asserting that A has a  
proof using large cardinals with at most 2^20 symbols, but not in ZFC  
using at most 2^2^2^20 symbols.

There is a kind of systematic idea here, coming out of my experience  
with Boolean Relation Theory. There, I look at reasonably natural  
*collections* of statements, much more natural than any individual one  
in the collection, and show that only one statement in the collection  
is independent of ZFC. Thus the overwhelming majority are decided in  
ZFC.

>
>
> In the past you have remarked that after proving several instances  
> of the
> wild-conjecture template, you started to develop a "feel" for when  
> there
> was a large cardinal.  Can you articulate that feeling in more detail?
> Failing that, can you give some examples of "TAME CONJECTURES"?   
> That is,
> take some unsolved problem X, and make the
>
> TAME CONJECTURE.  Either X is not provable even using large cardinals,
> or X is provable in ZFC.
>
> Are there any TAME CONJECTURES that you believe with roughly the same
> conviction as you believe your WILD CONJECTURES?
>
>

I have thought some about finding a proposed single reasonably  
concrete sentence which I think is likely NOT to be solved using large  
cardinals. But I don't have a good candidate for this. So I guess the  
answer to your quesiton is No.

Of course, we know that there exists a Turing machine such that the  
halting problem for it, is independent of large cardinals - assuming  
that we envision a finite or even r.e. set of large cardinal axioms.

In fact, we can say that among the TMs of size at most n, at least one  
has a halting problem is independent of ZFC. But it is not clear to me  
roughly how small we can make n for this purpose.

In fact, I have never seen a decent n offered with the claim that  
there is a TM of size n whose halting problem is independent of ZFC.

Harvey Friedman