Elementary function arithmetic: Difference between revisions

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Revision as of 21:55, 24 August 2022

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==Definition==

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

*two constants 0, 1,

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==Friedman's grand conjecture<!--'Friedman's grand conjecture' redirects here-->==

[[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.

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*'''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 Π{{su|p=0|b=2}} 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.

== See also ==

<!-- *[[ELEMENTARY]], a related computational complexity class - already wikilinked in the text as <nowiki>[[ELEMENTARY#Definition|bounded sums and products]]</nowiki> -->

* {{annotated link|Elementary function}}

*[[ {{annotated link|Grzegorczyk hierarchy]]}}

*[[ {{annotated link|Reverse mathematics]]}}

*[[ {{annotated link|Ordinal analysis]]}}

*[[ {{annotated link|Tarski's high school algebra problem]]}}

==References==

*{{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}}

{{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}}

{{Mathematical logic}}

[[Category:Conjectures]]

[[Category:Formal theories of arithmetic]]

[[Category:Proof theory]]

[[Category:Conjectures]]