In proof theory, a branch of mathematical logic, elementary function arithmetic or exponential function arithmetic (EFA) is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, xy, together with induction for formulas with bounded quantifiers.
EFA is a very weak logical system, whose proof theoretic ordinal is ω3, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic.
There are weak fragments of second-order arithmetic called RCA*
0 and WKL*
0 that have the same consistency strength as EFA and are conservative over it for Π0
2 sentences, which are sometimes studied in reverse mathematics.
Friedman's grand conjecture
Harvey Friedman's grand conjecture implies that many mathematical theorems, such as Fermat's last theorem, can be proved in very weak systems such as EFA.
The original statement of the conjecture from Friedman (1999) is:
- "Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of Peano Arithmetic based on the usual quantifier-free axioms for 0, 1, +, ×, exp, together with the scheme of induction for all formulas in the language all of whose quantifiers are bounded."
While it is easy to construct artificial arithmetical statements that are try but not provable in EFA, the point of Friedman's conjecture is that natural examples of such statements in mathematics seem to be rare. Some natural examples include consistency statements from logic, several statements related to Ramsey theory such as Szemeredi's lemma and the graph minor theorem, and Tarjan's algorithm for the disjoint-set data structure.
See also
- ELEMENTARY, a related computational complexity class
- Grzegorczyk hierarchy
- Reverse mathematics
- Tarski's high school algebra problem
References
- Avigad, Jeremy (2003), "Number theory and elementary arithmetic", Philosophia Mathematica. Philosophy of Mathematics, its Learning, and its Application. Series III, 11 (3): 257–284, doi:10.1093/philmat/11.3.257, ISSN 0031-8019, MR 2006194
- Friedman, Harvey (1999), grand conjectures