Elementary function arithmetic

Short description: System of arithmetic in proof theory

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"Elementary recursive arithmetic" redirects here. For the computational complexity class, see Elementary recursive function.

In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic,[1] 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.

## Contents

* 1 Definition
* 2 Friedman's grand conjecture
* 3 Related systems
* 4 See also
* 5 References

## Definition

EFA is a system in first-order logic (with equality). Its language contains:

* two constants 0, 1,
* three binary operations +, ×, exp, with exp(x,y) usually written as xy,
* a binary relation symbol < (This is not really necessary as it can be written in terms of the other operations and is sometimes omitted, but is convenient for defining bounded quantifiers).

Bounded quantifiers are those of the form ∀(x<y) and ∃(x<y) which are abbreviations for ∀x(x<y)->… and ∃x(x<y)∧… in the usual way.

The axioms of EFA are

* The axioms of Robinson arithmetic for 0, 1, +, ×, <
* The axioms for exponentiation: x0=1, xy+1=xy×x.
* Induction for formulas all of whose quantifiers are bounded (but which may contain free variables).

## 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 true 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 the Szemerédi regularity lemma, and the graph minor theorem.

## Related systems

Several related computational complexity classes have similar properties to EFA:

* One can omit the binary function symbol exp from the language, by taking Robinson arithmetic together with induction for all formulas with bounded quantifiers and an axiom stating roughly that exponentiation is a function defined everywhere. This is similar to EFA and has the same proof-theoretic strength, but is more cumbersome to work with.
* There are weak fragments of second-order arithmetic called RCA0* and WKL0* that are conservative over EFA for Π20 sentences (i.e. any Π20 sentences proven by RCA0* or WKL0* are already proven by EFA.)[2] In particular, they are conservative for consistency statements. These fragments are sometimes studied in reverse mathematics (Simpson 2009).
* Elementary recursive arithmetic (ERA) is a subsystem of primitive recursive arithmetic (PRA) in which recursion is restricted to bounded sums and products. This also has the same Π20 sentences as EFA, in the sense that whenever EFA proves ∀x∃y P(x,y), with P quantifier-free, ERA proves the open formula P(x,T(x)), with T a term definable in ERA. Like PRA, ERA can be defined in an entirely logic-free[clarification needed] manner, with just the rules of substitution and induction, and defining equations for all elementary recursive functions. Unlike PRA, however, the elementary recursive functions can be characterized by the closure under composition and projection of a finite number of basis functions, and thus only a finite number of defining equations are needed.

## See also

* Elementary function - Type of mathematical function
* Grzegorczyk hierarchy - Functions in computability theory
* Reverse mathematics - Branch of mathematical logic
* Ordinal analysis - Mathematical technique used in proof theory
* Organization:Tarski's high school algebra problem - Mathematical problem

## References

1. ↑ C. Smoryński, "Nonstandard Models and Related Developments" (p. 217). From Harvey Friedman's Research on the Foundations of Mathematics (1985), Studies in Logic and the Foundations of Mathematics vol. 117.
2. ↑ S. G. Simpson, R. L. Smith, "Factorization of polynomials and Σ10-induction" (1986). Annals of Pure and Applied Logic, vol. 31 (p.305)

* Avigad, Jeremy (2003), "Number theory and elementary arithmetic", Philosophia Mathematica, Series III 11 (3): 257–284, doi:10.1093/philmat/11.3.257, ISSN 0031-8019
* Friedman, Harvey (1999), grand conjectures, http://cs.nyu.edu/pipermail/fom/1999-April/003014.html
* Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, ISBN 978-0-521-88439-6, http://www.math.psu.edu/simpson/sosoa/

* v
* t
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Mathematical logic

General|

* Formal language
* Formation rule
* Formal proof
* Formal semantics
* Well-formed formula
* Set
* Element
* Class
* Classical logic
* Axiom
* Rule of inference
* Relation
* Theorem
* Logical consequence
* Type theory
* Symbol
* Syntax
* Theory

Systems|

* Formal system
* Deductive system
* Axiomatic system
* Hilbert style systems
* Natural deduction
* Sequent calculus

Traditional logic|

* Proposition
* Inference
* Argument
* Validity
* Cogency
* Syllogism
* Square of opposition
* Venn diagram

Propositional calculus
and Boolean logic

* Boolean functions
* Propositional calculus
* Propositional formula
* Logical connectives
* Truth tables
* Many-valued logic

Predicate logic|

* First-order
* Quantifiers
* Predicate
* Second-order
* Monadic predicate calculus

Naive set theory|

* Set
* Empty set
* Element
* Enumeration
* Extensionality
* Finite set
* Infinite set
* Subset
* Power set
* Countable set
* Uncountable set
* Recursive set
* Domain
* Codomain
* Image
* Map
* Function
* Relation
* Ordered pair

Set theory|

* Foundations of mathematics
* Zermelo–Fraenkel set theory
* Axiom of choice
* General set theory
* Kripke–Platek set theory
* Von Neumann–Bernays–Gödel set theory
* Morse–Kelley set theory
* Tarski–Grothendieck set theory

Model theory|

* Model
* Interpretation
* Non-standard model
* Finite model theory
* Truth value
* Validity

Proof theory|

* Formal proof
* Deductive system
* Formal system
* Theorem
* Logical consequence
* Rule of inference
* Syntax

Computability
theory|

* Recursion
* Recursive set
* Recursively enumerable set
* Decision problem
* Church–Turing thesis
* Computable function
* Primitive recursive function

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