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view source for second order arithmetic wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in view source for second order arithmetic add languages article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file page information get shortened url switch to legacy parser in other projects wikidata item appearance move to sidebar hide second order arithmetic you do not have permission to edit this page for the following reason this ip address range has been globally blocked this does not affect your ability to read wikipedia pages most people who see this message have done nothing wrong some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism and can sometimes affect other users who are unrelated to that abuse review the information below for assistance if you do not believe that you have done anything wrong this block affects editing on all wikimedia wikis the ip address or range 5 135 0 0 16 has been globally blocked by jon kolbert for the following reason s open proxy webhost visit the faq if you are affected this block will expire on 04 29 8 january 2027 your current ip address is 5 135 42 194 even while globally blocked you will usually still be able to edit pages on meta wiki if you believe you were blocked by mistake you can find additional information and instructions in the stewards block wizard other useful links global blocks help i have been blocked you can view and copy the source of this page axioms basic 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 axioms peano arithmetic as first order theory peano arithmetic without induction the domain of discourse for the quantification logic quantified variable s is the natural number s collectively denoted by n and including the distinguished member math 0 math called zero the primitive functions are the unary successor function denoted by prefix math s math and two binary operation s addition and multiplication denoted by the infix operator and math cdot math respectively there is also a primitive binary relation called order relation order denoted by the infix operator axioms governing the successor function and zero math forall m sm 0 rightarrow bot math the successor of a natural number is never zero math forall m forall n sm sn rightarrow m n math the successor function is injective function injective math forall n 0 n lor exists m sm n math every natural number is zero or a successor addition defined recursion recursively li value 4 math forall m m 0 m math math forall m forall n m sn s m n math multiplication defined recursively li value 6 math forall m m cdot 0 0 math math forall m forall n m cdot sn m cdot n m math axioms governing the order relation li value 8 math forall m m 0 rightarrow bot math no natural number is smaller than zero math forall n forall m m sn leftrightarrow m n lor m n math math forall n 0 n lor 0 n math every natural number is zero or bigger than zero math forall m forall n sm n lor sm n leftrightarrow m n math these axioms are all first order logic first order statements that is all variables range over the natural number s and not set theory 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 peano axioms axiom schema of induction make up the usual peano axioms peano dedekind definition of n adding to these axioms any sort of peano axioms axiom schema of induction makes redundant the axioms 3 10 and 11 induction and comprehension schema 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 sub 1 sub m sub k sub and x sub 1 sub x sub l sub the induction axiom for φ is the axiom math 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 math 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 math n in x math 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 math forall x 0 in x land forall n n in x rightarrow sn in x rightarrow forall n n in x math 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 math exists z forall n n in z leftrightarrow varphi n math this axiom makes it possible to form the set math z n varphi n math 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 math n not in z math would lead to the comprehension axiom math exists z forall n n in z leftrightarrow n not in z math which is inconsistent this convention is assumed in the remainder of this article return to second order arithmetic retrieved from https en wikipedia org wiki second order_arithmetic privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search view source for second order arithmetic add languages add topic