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tic is denoted by z 2 second order arithmetic includes but is significantly stronger than its first order counterpart peano arithmetic unlike peano arithmetic second order arithmetic allows quantification over sets of natural numbers as well as numbers themselves because real numbers can be represented as infinite sets of natural numbers in well known ways and because second order arithmetic allows quantification over such sets it is possible to formalize the real numbers in second order arithmetic for this reason second order arithmetic is sometimes called analysis 2 second order arithmetic can also be seen as a weak version of set theory in which every element is either a natural number or a set of natural numbers although it is much weaker than zermelo fraenkel set theory second order arithmetic can prove essentially all of the results of classical mathematics expressible in its language a subsystem of second order arithmetic is a theory in the language of second order arithmetic each axiom of which is a theorem of full second order arithmetic z 2 such subsystems are essential to reverse mathematics a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength much of core mathematics can be formalized in these weak subsystems some of which are defined below reverse mathematics also clarifies the extent and manner in which classical mathematics is nonconstructive definition edit syntax edit the language of second order arithmetic is two sorted the first sort of terms and in particular variables usually denoted by lower case letters consists of individuals whose intended interpretation is as natural numbers the other sort of variables variously called set variables class variables or even predicates are usually denoted by upper case letters they refer to classes predicates properties of individuals and so can be thought of as sets of natural numbers both individuals and set variables can be quantified universally or existentially a formula with no bound set variables that is no quantifiers over set variables is called arithmetical an arithmetical formula may have free set variables and bound individual variables individual terms are formed from the constant 0 the unary function s the successor function and the binary operations and displaystyle cdot addition and multiplication the successor function adds 1 to its input the relations equality and comparison of natural numbers relate two individuals whereas the relation membership relates an individual and a set or class thus in notation the language of second order arithmetic is given by the signature l 0 s displaystyle mathcal l 0 s cdot in for example n n x s n x displaystyle forall n n in x rightarrow sn in x is a well formed formula of second order arithmetic that is arithmetical has one free set variable x and one bound individual variable n but no bound set variables as is required of an arithmetical formula whereas x n n x n s s s s s s 0 s s s s s s s 0 displaystyle exists x forall n n in x leftrightarrow n ssssss0 cdot sssssss0 is a well formed formula that is not arithmetical having one bound set variable x and one bound individual variable n semantics edit several different interpretations of the quantifiers are possible if second order arithmetic is studied using the full semantics of second order logic then the set quantifiers range over all subsets of the range of the individual variables if second order arithmetic is formalized using the semantics of first order logic henkin semantics then any model includes a domain for the set variables to range over and this domain may be a proper subset of the full powerset of the domain of individual variables 3 axioms edit basic edit the following axioms are known as the basic axioms or sometimes the robinson axioms the resulting first order theory known as robinson arithmetic is essentially peano arithmetic without induction the domain of discourse for the quantified variables is the natural numbers collectively denoted by n and including the distinguished member 0 displaystyle 0 called zero the primitive functions are the unary successor function denoted by prefix s displaystyle s and two binary operations addition and multiplication denoted by the infix operator and displaystyle cdot respectively there is also a primitive binary relation called order denoted by the infix operator axioms governing the successor function and zero m s m 0 displaystyle forall m sm 0 rightarrow bot the successor of a natural number is never zero m n s m s n m n displaystyle forall m forall n sm sn rightarrow m n the successor function is injective n 0 n m s m n displaystyle forall n 0 n lor exists m sm n every natural number is zero or a successor addition defined recursively m m 0 m displaystyle forall m m 0 m m n m s n s m n displaystyle forall m forall n m sn s m n multiplication defined recursively m m 0 0 displaystyle forall m m cdot 0 0 m n m s n m n m displaystyle forall m forall n m cdot sn m cdot n m axioms governing the order relation m m 0 displaystyle forall m m 0 rightarrow bot no natural number is smaller than zero n m m s n m n m n displaystyle forall n forall m m sn leftrightarrow m n lor m n n 0 n 0 n displaystyle forall n 0 n lor 0 n every natural number is zero or bigger than zero m n s m n s m n m n displaystyle forall m forall n sm n lor sm n leftrightarrow m n these axioms are all first order statements that is all variables range over the natural numbers and not sets thereof a fact even stronger than their being arithmetical moreover there is but one existential quantifier in axiom 3 axioms 1 and 2 together with an axiom schema of induction make up the usual peano dedekind definition of n adding to these axioms any sort of axiom schema of induction makes redundant the axioms 3 10 and 11 induction and comprehension schema edit if φ n is a formula of second order arithmetic with a free individual variable n and possibly other free individual or set variables written m 1 m k and x 1 x l the induction axiom for φ is the axiom m 1 m k x 1 x l φ 0 n φ n φ s n n φ n displaystyle forall m_ 1 dots m_ k forall x_ 1 dots x_ l varphi 0 land forall n varphi n rightarrow varphi sn rightarrow forall n varphi n the full second order induction scheme consists of all instances of this axiom over all second order formulas one particularly important instance of the induction scheme is when φ is the formula n x displaystyle n in x expressing the fact that n is a member of x x being a free set variable in this case the induction axiom for φ is x 0 x n n x s n x n n x displaystyle forall x 0 in x land forall n n in x rightarrow sn in x rightarrow forall n n in x this sentence is called the second order induction axiom if φ n is a formula with a free variable n and possibly other free variables but not the variable z the comprehension axiom for φ is the formula z n n z φ n displaystyle exists z forall n n in z leftrightarrow varphi n this axiom makes it possible to form the set z n φ n displaystyle z n varphi n of natural numbers satisfying φ n there is a technical restriction that the formula φ may not contain the variable z for otherwise the formula n z displaystyle n not in z would lead to the comprehension axiom z n n z n z displaystyle exists z forall n n in z leftrightarrow n not in z which is inconsistent this convention is assumed in the remainder of this article the full system edit the formal theory of second order arithmetic in the language of second order arithmetic consists of the basic axioms the comprehension axiom for every formula φ arithmetic or otherwise and the second order induction 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 display...
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