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n axiom this theory is sometimes called full second order arithmetic to distinguish it from its subsystems defined below because full second order semantics imply that every possible set exists the comprehension axioms may be taken to be part of the deductive system when full second order semantics is employed 3 models edit this section describes second order arithmetic with first order semantics thus a model m displaystyle mathcal m of the language of second order arithmetic consists of a set m which forms the range of individual variables together with a constant 0 an element of m a function s from m to m two binary operations and on m a binary relation on m and a collection d of subsets of m which is the range of the set variables omitting d produces a model of the language of first order arithmetic when d is the full powerset of m the model m displaystyle mathcal m is called a full model the use of full second order semantics is equivalent to limiting the models of second order arithmetic to the full models in fact the axioms of second order arithmetic have only one full model this follows from the fact that the peano axioms with the second order induction axiom have only one model under second order semantics definable functions edit the first order functions that are provably total in second order arithmetic are precisely the same as those representable in system f 4 almost equivalently system f is the theory of functionals corresponding to second order arithmetic in a manner parallel to how gödel s system t corresponds to first order arithmetic in the dialectica interpretation more types of models edit when a model of the language of second order arithmetic has certain properties it can also be called these other names when m is the usual set of natural numbers with its usual operations m displaystyle mathcal m is called an ω model in this case the model may be identified with d its collection of sets of naturals because this set is enough to completely determine an ω model the unique full ω displaystyle omega model which is the usual set of natural numbers with its usual structure and all its subsets is called the intended or standard model of second order arithmetic 5 a model m displaystyle mathcal m of the language of second order arithmetic is called a β model if m 1 1 p ω displaystyle mathcal m prec _ 1 1 mathcal p omega i e the σ 1 1 statements with parameters from m displaystyle mathcal m that are satisfied by m displaystyle mathcal m are the same as those satisfied by the full model 6 some notions that are absolute with respect to β models include a ω ω displaystyle a subseteq omega times omega encodes a well order 7 and a ω ω displaystyle a subseteq omega times omega is a tree 6 the above result has been extended to the concept of a β n model for n n displaystyle n in mathbb n which has the same definition as the above except 1 1 displaystyle prec _ 1 1 is replaced by n 1 displaystyle prec _ n 1 i e σ 1 1 displaystyle sigma _ 1 1 is replaced by σ n 1 displaystyle sigma _ n 1 6 using this definition β 0 models are the same as ω models 8 subsystems edit main article reverse mathematics there are many named subsystems of second order arithmetic a subscript 0 in the name of a subsystem indicates that it includes only a restricted portion of the full second order induction scheme 9 such a restriction lowers the proof theoretic strength of the system significantly for example the system aca 0 described below is equiconsistent with peano arithmetic the corresponding theory aca consisting of aca 0 plus the full second order induction scheme is stronger than peano arithmetic arithmetical comprehension edit many of the well studied subsystems are related to closure properties of models for example it can be shown that every ω model of full second order arithmetic is closed under turing jump but not every ω model closed under turing jump is a model of full second order arithmetic the subsystem aca 0 includes just enough axioms to capture the notion of closure under turing jump aca 0 is defined as the theory consisting of the basic axioms the arithmetical comprehension axiom scheme in other words the comprehension axiom for every arithmetical formula φ and the ordinary second order induction axiom it would be equivalent to also include the entire arithmetical induction axiom scheme in other words to include the induction axiom for every arithmetical formula φ it can be shown that a collection s of subsets of ω determines an ω model of aca 0 if and only if s is closed under turing jump turing reducibility and turing join 10 the subscript 0 in aca 0 indicates that not every instance of the induction axiom scheme is included this subsystem this makes no difference for ω models which automatically satisfy every instance of the induction axiom it is of importance however in the study of non ω models the system consisting of aca 0 plus induction for all formulas is sometimes called aca with no subscript the system aca 0 is a conservative extension of first order arithmetic or first order peano axioms defined as the basic axioms plus the first order induction axiom scheme for all formulas φ involving no class variables at all bound or otherwise in the language of first order arithmetic which does not permit class variables at all in particular it has the same proof theoretic ordinal ε 0 as first order arithmetic owing to the limited induction schema the arithmetical hierarchy for formulas edit main article arithmetical hierarchy a formula is called bounded arithmetical or δ 0 0 when all its quantifiers are of the form n t or n t where n is the individual variable being quantified and t is an individual term where n t displaystyle forall n t cdots stands for n n t displaystyle forall n n t rightarrow cdots and n t displaystyle exists n t cdots stands for n n t displaystyle exists n n t land cdots a formula is called σ 0 1 or sometimes σ 1 respectively π 0 1 or sometimes π 1 when it is of the form mφ respectively mφ where φ is a bounded arithmetical formula and m is an individual variable that is free in φ more generally a formula is called σ 0 n respectively π 0 n when it is obtained by adding existential respectively universal individual quantifiers to a π 0 n 1 respectively σ 0 n 1 formula and σ 0 0 and π 0 0 are both equal to δ 0 0 by construction all these formulas are arithmetical no class variables are ever bound and in fact by putting the formula in skolem prenex form one can see that every arithmetical formula is logically equivalent to a σ 0 n or π 0 n formula for all large enough n recursive comprehension edit the subsystem rca 0 is a weaker system than aca 0 and is often used as the base system in reverse mathematics it consists of the basic axioms the σ 0 1 induction scheme and the δ 0 1 comprehension scheme the former term is clear the σ 0 1 induction scheme is the induction axiom for every σ 0 1 formula φ the term δ 0 1 comprehension is more complex because there is no such thing as a δ 0 1 formula the δ 0 1 comprehension scheme instead asserts the comprehension axiom for every σ 0 1 formula that is logically equivalent to a π 0 1 formula this scheme includes for every σ 0 1 formula φ and every π 0 1 formula ψ the axiom m x n φ n ψ n z n n z φ n displaystyle forall m forall x forall n varphi n leftrightarrow psi n rightarrow exists z forall n n in z leftrightarrow varphi n the set of first order consequences of rca 0 is the same as those of the subsystem i σ 1 of peano arithmetic in which induction is restricted to σ 0 1 formulas in turn i σ 1 is conservative over primitive recursive arithmetic pra for π 2 0 displaystyle pi _ 2 0 sentences moreover the proof theoretic ordinal of r c a 0 displaystyle mathrm rca _ 0 is ω ω the same as that of pra it can be seen that a collection s of subsets of ω determines an ω model of rca 0 if and only if s is closed under turing reducibility and turing join in particular the collection of all computable subsets of ω gives an ω model of rca 0 this is the motivation behind the name of this system if a set can be proved to exist using rca 0 then the set is recursive i e computable weaker systems edit sometimes an even weaker system than rca 0 is desired one such system is defined as follows one must first augment the language of arithmetic with an exponential function symbol in stronger systems the exponential can be defined in terms of addition and multiplication by the usual trick but when the system becomes too weak this is no longer possible and the basic axioms by the obvious axioms defining exponentiation inductively from multiplication then the system consists of the enriched basic axioms plus δ 0 1 comprehension plus δ 0 0 induction stronger systems edit over aca 0 each formula of second order arithmetic is equivalent to a σ 1 n or π 1 n formula for all large enough n the system π 1 1 comprehension is the system consisting of the basic axioms plus the ordinary second order induction axiom and the comprehension axiom for every boldface 11 π 1 1 formula φ this is equivalent to σ 1 1 comprehension on the other hand δ 1 1 comprehension defined analogously to δ 0 1 comprehension is weaker projective determinacy edit main article axiom of projective determinacy projective determinacy is the assertion that every two player perfect information game with moves being natural numbers game length ω and projective payoff set is determined that is one of the players has a winning strategy the first player wins the game if the play belongs to the payoff set otherwise the second player wins a set is projective if and only if as a predicate it is expressible by a formula in the language of second order arithmetic allowing real numbers as parameters so projective determinacy is expressible as a schema in the language of z 2 many natural propositions expressible in the language of second order arithmetic are independent of z 2 and even zfc but are provable from projective determinacy examples include coanalytic perfect subset property measurability and the property of baire for σ 2 1 displaystyle sigma _ 2 1 sets π 3 1 displaystyle pi _ 3 1 uniformization etc over a weak base theory such as rca 0 projective determinacy implies comprehension and provides an essentially complete theory of second order arithmetic natural statements in the language of z 2 that are independent of z 2 with projective determinacy are hard to find 12 zfc there are n woodin cardinals n is a natural number is conservative over z 2 with projective determinacy citation needed that is a statement in the language of second order arithmetic is provable in z 2 with projective determinacy if and only if its translation into the language of set theory is provable in zfc there are n woodin cardinals n n coding mathematics edit second order arithmetic directly formalizes natural numbers and sets of natural numbers however it is able to formalize other mathematical objects indirectly via coding techniques a fact that was first noticed by weyl 13 the integers rational numbers and real numbers can all be formalized in the subsystem rca 0 along with complete separable metric spaces and continuous functions between them 14 the research program of reverse mathematics uses these formalizations of mathematics in second order arithmetic to study the set existence axioms required to prove mathematical theorems 15 for example the intermediate value theorem for functions from the reals to the reals is provable in rca 0 16 while the bolzano weierstrass theorem is equivalent to aca 0 over rca 0 17 the aforementioned coding works well for continuous and total functions assuming a higher order base theory plus weak kőnig s lemma 18 as perhaps expected in the case of topology coding is not without problems 19 see also edit paris harrington theorem presburger arithmetic true arithmetic references edit hilbert d bernays p 1934 grundlagen der mathematik springer verlag mr 0237246 sieg w 2013 hilbert s programs and beyond oxford university press p 291 isbn 978 0 19 970715 7 1 2 shapiro stewart 1991 foundations without foundationalism a case for second order logic oxford logic guides vol 17 the clarendon press oxford university press new york pp 66 74 75 isbn 0 19 853391 8 mr 1143781 girard jean yves 1987 proofs and types translated by taylor paul cambridge university press pp 122 123 isbn 0 521 37181 3 simpson s g 2009 subsystems of second order arithmetic perspectives in logic 2nd ed cambridge university press pp 3 4 isbn 978 0 521 88439 6 mr 2517689 1 2 3 marek w 1974 1975 stable sets a characterization of β 2 models of full second order arithmetic and some related facts fundamenta mathematicae 82 175 189 doi 10 4064 fm 82 2 175 189 mr 0373897 marek w 1978 ω models of second order arithmetic and admissible sets fundamenta mathematicae 98 2 103 120 doi 10 4064 fm 98 2 103 120 mr 0476490 marek w 1973 observations concerning elementary extensions of ω models ii the journal of symbolic logic 38 227 231 doi 10 2307 2272059 jstor 2272059 mr 0337612 friedman h 1976 systems of second order arithmetic with restricted induction i ii meeting of the association for symbolic logic journal of symbolic logic abstracts 41 557 559 jstor 2272259 simpson 2009 pp 311 313 welch p d 2011 weak systems of determinacy and arithmetical quasi inductive definitions pdf the journal of symbolic logic 76 2 418 436 doi 10 2178 jsl 1305810756 mr 2830409 woodin w h 2001 the continuum hypothesis part i notices of the american mathematical society 48 6 simpson 2009 p 16 simpson 2009 chapter ii simpson 2009 p 32 simpson 2009 p 87 simpson 2009 p 34 kohlenbach ulrich 2002 foundational and mathematical uses of higher types reflections on the foundations of mathematics essays in honor of solomon feferman papers from the symposium held at stanford university stanford ca december 11 13 1998 lecture notes in logic vol 15 urbana illinois association for symbolic logic pp 92 116 isbn 1 56881 169 1 mr 1943304 hunter james 2008 higher order reverse topology pdf doctoral dissertation university of madison wisconsin further reading edit burgess j p 2005 fixing frege princeton university press buss s r 1998 handbook of proof theory elsevier isbn 0 444 89840 9 takeuti g 1975 proof theory north holland publishing company isbn 0 444 10492 5 retrieved from https en wikipedia org w index php title second order_arithmetic oldid 1368604233 category formal theories of arithmetic hidden categories articles with short description short description matches wikidata all articles with unsourced statements articles with unsourced statements from july 2023 this page was last edited on 10 august 2026 at 01 44 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia 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