[FOM] irrational conjectures

Harvey Friedman friedman at math.ohio-state.edu
Sat Mar 14 04:35:50 EDT 2009

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  

> 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

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