[FOM] logical complexity of mathematical statements

[Harvey Friedman]

On Sun, Oct 11, 2015 at 2:18 PM, Joe Shipman <joeshipman at aol.com> wrote:
> Schanuel's Conjecture states that if a1, a2, ... , a_n are complex numbers that are linearly independent over the rationals, then adjoining them and their exponentials to the rationals gives a field with transcendence degree at least n.
>
> Schanuel's conjecture has interesting arithmetical consequences, but is it equivalent to an arithmetical statement? If not, how low in the logical hierarchy of statements of second-order arithmetic can it or an equivalent statement be found?

Pi-1-1 is a crude upper bound for this sort of thing.

There is a good Wiki article
https://en.wikipedia.org/wiki/Schanuel%27s_conjecture

It includes some important material that I copy below that needs to be
followed up from the logical complexity point of view.

"The conjecture, if proven, would generalize most known results in
transcendental number theory. The special case where the numbers
z1,...,zn are all algebraic is the Lindemann–Weierstrass theorem."

It is easy to see that the Lindemann-Weierstrass statement is Pi02.
Now that it is a theorem it is of course Pi00. But the question is:
what is the bound of the Pi02 statement associated with it? This
question can be stated purely mathematically without logical
conversion to Pi02. This is done by talking about "the extent to which
real numbers are mutually transcendental",,in terms of the errors
involved in thinking that it is not the case.

What is not clear to me at the moment is the effect of moving to general reals.

"If, on the other hand, the numbers are chosen so as to make
exp(z1),...,exp(zn) all algebraic then one would prove that linearly
independent logarithms of algebraic numbers are algebraically
independent, a strengthening of Baker's theorem."

"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? Again,
probably with more trouble, one can set up a mathematical treatment of
just what bounds for the corresponding Pi03 statement means
mathematically.

"The Gelfond–Schneider theorem follows from this strengthened version
of Baker's theorem, as does the currently unproven four exponentials
conjecture."

The Gelfond-Schneider theorem says that

"if a and b are algebraic numbers with a ≠ 0,1 and b irrational, then
a^b is a transcendental number."

This is Pi02, and subject to the bound question stated mathematically.
What bound is given by the known proofs?

The four exponentials conjecture says that

"If x1, x2 and y1, y2 are two pairs of complex numbers, with each pair
being linearly independent over the rational numbers, then at least
one of the following four numbers is transcendental:

e^x1y1,   e^x1y2,   e^x2y1,   e^x2y2"

Again, just like in Schanuel's Conjecture, it's a new ball game when
we are talking about arbitrary reals (or complexes).

These statements are obviously Pi-1-1. The question is whether they
can be made arithmetical (or Sigma-1-1).

It would seem likely that the only way that Schanuel's Conjecture can
be shown to be arithmetical (or even Sigma-1-1) would be to use
delicate probably unknown facts about exponentiation.

The right way to start going about clarifying this situation is to
look at the statement

*for all reals x,y linearly independent over the rationals, f(x),f(y)
are mutually transcendental"

where f is an exponential time function from R into R, like exponentiation.

CONJECTURE. For every Pi-1-1 sentence A there is an exponential time
f:R into R such that *) above is provably equivalent to A over RCA_0.

"It is also known that Schanuel's conjecture would be a consequence of
conjectural results in the theory of motives. There Grothendieck's
period conjecture for an abelian variety A states that the
transcendence degree of its period matrix is the same as the dimension
of the associated Mumford–Tate group, and what is known by work of
Pierre Deligne is that the dimension is an upper bound for the
transcendence degree. Bertolin has shown how a generalised period
conjecture includes Schanuel's conjecture.[7]"

All of these items should also be looked at through the lens of
logical complexity.

I still adhere to the general principle that, roughly speaking,

***all mathematics is Pi01, except for higher statements that, when
fully solved, will reveal new Pi01 information that implies the
original statement using known methods***.

*** would apply to all Pi02 and Pi03 theorems. They are directly
begging to become Pi01. When a really interesting mathematical
statement is proved, generally speaking there is some sort of new
underlying Pi01, Pi02, or Pi03 fact. It would be interesting to know
just how well *** holds up for mainstream mathematical developments.

Harvey Friedman