Summary: Update procedures and the
1-consistency of arithmetic
February 8, 2002
The 1-consistency of arithmetic is shown to be equivalent to the ex-
istence of fixed points of a certain type of update procedure, which is
implicit in the epsilon-substitution method.
A theory in the language of arithmetic is said to be 1-consistent if it is consistent
with every true universal sentence, or, equivalently, if every existential sentence
it proves is true. The main theorem in this paper asserts that the 1-consistency
of first-order Peano arithmetic is equivalent, over a weak metatheory, to the
assertion that one can always solve certain systems of equations involving finite
partial functions on the natural numbers.
It has been noted before (for example, by Tait, in ) that one can view
Hilbert's epsilon-substitution method as posing the problem of finding solutions
to systems of equations of a certain kind. The theorem just mentioned simply
characterizes the types of equations that need to be solved, in a way that, I
hope, is intuitive and helps clarify the computational content of the problem.