[FOM] logical complexity of mathematical statements

[joeshipman at aol.com]

As stated, this does NOT QUITE convert Baker's theorem into a Pi01 sentence, because there is no independent way given to test that the logarithms are linearly independent. If you are given algebraic numbers a0, a1, ..., a_n and b1,b2,...,bn and you find that 
a0 + a1log(b1) + a2log(b2) + ... + a_n log(b_n) is closer to 0 than the effective lower bound given by Baker, you haven't refuted his statement, you've just shown that the logs of the b_i weren't linearly independent after all.


Can this be gotten around and a genuine Pi01 sentence be extracted from it?


-- JS



-----Original Message-----
From: Timothy Y. Chow <tchow at alum.mit.edu>
To: fom <fom at cs.nyu.edu>
Sent: Wed, Oct 14, 2015 1:12 pm
Subject: Re: [FOM] logical complexity of mathematical statements


Harvey Friedman wrote:

> "Linearly independent logarithms of algebraic
numbers are
> algebraically independent" by similar reasoning seems to be Pi03.
Look
> at Baker's Theorem, a special case. Does Baker give a bound for the
>
corresponding Pi03 statement, and if so, what kind of bound?

Yes, if L_1,
..., L_n are the linearly independent logarithms of algebraic 
numbers and a_0,
..., a_n are algebraic numbers (not all zero) then Baker 
gives an effectively
computable nonzero lower bound for

   | a_0 + a_1 L_1 + ... + a_n L_n
|

Tim
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