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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 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 arithme...
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