# [FOM] Maximal Principle in Arithmetic

Harvey Friedman friedman at math.ohio-state.edu
Mon Jun 2 09:39:06 EDT 2003

```I will be discussing the following result in an upcoming seminar.
Assuming all is well, I will make a more elaborate numbered posting

We are in the language of ordered rings, 0,1,+,x,<.

Consider the following three axioms.

1. DOR. Discrete ordered ring axioms.
2. CM. Common multiple axiom. For all n >= 1 there exists m >= 1 such
that m is a common multiple of 1,...,n.
3. MAX(4,2). Any value of any quadratic polynomial in four variables
is contained in a maximal interval (possible infinite) of such values.

Or these.

1. DOR.
2. CM.
3. RMAX(4,2). Any value of any quadratic polynomial in four variables
whose arguments are restricted to a single finite interval, is
contained in a maximal interval of such values.

The result is that

1) DOR + CM + MAX(4,2) implies IDelta0 + exp.

2) DOR + CM + RMAX(4,2) is equivalent to IDelta0 + exp.

3) It is morally certain that DOR + CM + MAX(4,2) is equivalent to
IDelta0 + exp, but it will take some detailed checking of the known
solution to Hilbert's tenth problem for quadratic polynomials.
```