[FOM] irrational conjectures

[Timothy Y. Chow]

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?

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?

Tim