Re: Independent statements?

> 2. Do there exist a couple of statements T1, T2 which are proved to be
> either  true or false within some axiomatic system X while no proofs
> are currently available to judge the final fates of T1 and T2? If yes,
> go on.

Yes.   Let n be some VERY large number (perhaps 10^10^10 is enough, or else
take Ackermann(100,100).   Let T1 be the statement 

"Goldbach's conjecture (or some other currently open Pi_1 statement -- 
 you have to be a bit careful how to define "open"...) 
 is true up to n, i.e., all even numbers up to n are the sum of
 2 primes". 

Then T1 is decidable in any reasonable system X (such as X=Peano arithmetic,
or X=ZFC), i.e.,  X will either prove T1 or refute T1.  (because T1
uses only bounded quantifiers.)   
But no proof of T1 is currently known, and also no proof of "non-T1". 

> 3. If T1 and T2 are the statements satisfying 2. We'll define a
> phrase:
>  'T1 and T2 are independent within X' if and only if none of the
> following 4 statements holds within X,
> a)(T1 is true)=>(T2 is true)
> b)(T1 is true)=>(T2 is false)

If T2 is either provable or refutable in X, then either a) or b) are 
provable in X, though we may not know which one.  
(If T1 is refutable in X, then in fact both a) and b) are provable in X)

> c)(T1 is false)=>(T2 is true)
> d)(T1 is false)=>(T2 is false)


In fact, for any system X (that includes reasoning in classical logic),
X can prove that exactly 3 of the 4 statements a,b,c,d are true. 

If T1 and T2 are as in 2 (i.e., X proves or refutes T1, and proves
or refutes T2), then X proves exactly 3 of the 4 statements a,b,c,d, 
and refutes the fourth.
So your formula "T1 and T2 are independent within X" is never true
if they satisfy 2. 
(Assuming I interpreted you correctly, i.e. "true in X" means "provable
in X")


Martin Goldstern