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==Friedman's grand conjecture<!--'Friedman's grand conjecture' redirects here-->==
[[Harvey Friedman (mathematician)|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 {{harvtxt|Friedman|1999}} is:
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: "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 [[mathematical induction|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==
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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 <math>\mathsf{RCA}_0^*</math> and <math>\mathsf{WKL}_0^*</math> that are conservative over EFA for <math>\Pi_2^0</math> sentences (i.e. any <math>\Pi_2^0</math> sentences proven by <math>\mathsf{RCA}_0^*</math> or <math>\mathsf{WKL}_0^*</math> are already proven by EFA.)<ref>S. G. Simpson, R. L. Smith, "[https://www.sciencedirect.com/science/article/pii/0168007286900746 Factorization of polynomials and <math>\Sigma_1^0</math>-induction]" (1986). [[Annals of Pure and Applied Logic]], vol. 31 (p.305)</ref> In particular, they are conservative for consistency statements. These fragments are sometimes studied in [[reverse mathematics]] {{harv|Simpson|2009}}.
*'''Elementary recursive arithmetic''' ('''ERA''') is a subsystem of [[primitive recursive arithmetic]] (PRA) in which recursion is restricted to [[ELEMENTARY#Definition|bounded sums and products]]. This also has the same <math>\Pi_2^0</math> 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{{clarify|date=November 2017}} 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.
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{{reflist}}
* {{Citation | last1=Avigad | first1=Jeremy | title=Number theory and elementary arithmetic | doi=10.1093/philmat/11.3.257 | mr=2006194 | year=2003 | journal=[[Philosophia Mathematica]] |series=Series III | issn=0031-8019 | volume=11 | issue=3 | pages=257–284}}
* {{citation |first=Harvey |last=Friedman |title=grand conjectures |year=1999 |url=http://cs.nyu.edu/pipermail/fom/1999-April/003014.html}}
* {{Citation | last1=Simpson | first1=Stephen G. |authorlink=Steve Simpson (mathematician)| title=Subsystems of second order arithmetic | url=http://www.math.psu.edu/simpson/sosoa/ | publisher=[[Cambridge University Press]] | edition=2nd | series=Perspectives in Logic | isbn=978-0-521-88439-6 | mr=1723993 | year=2009}}
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