Models Peano axioms

1 models

1.1 nonstandard models
1.2 set-theoretic models
1.3 interpretation in category theory

models

a model of peano axioms triple (n, 0, s), n (necessarily infinite) set, 0 ∈ n , s : n → n satisfies axioms above. dedekind proved in 1888 book, nature , meaning of numbers (german: sind und sollen die zahlen?, i.e., “what numbers , should be?”) 2 models of peano axioms (including second-order induction axiom) isomorphic. in particular, given 2 models (na, 0a, sa) , (nb, 0b, sb) of peano axioms, there unique homomorphism f : na → nb satisfying

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{\displaystyle {\begin{aligned}f(0_{a})&=0_{b}\\f(s_{a}(n))&=s_{b}(f(n))\end{aligned}}}

and bijection. means second-order peano axioms categorical. not case first-order reformulation of peano axioms, however.

nonstandard models

although usual natural numbers satisfy axioms of pa, there other models (called non-standard models); compactness theorem implies existence of nonstandard elements cannot excluded in first-order logic. upward löwenheim–skolem theorem shows there nonstandard models of pa of infinite cardinalities. not case original (second-order) peano axioms, have 1 model, isomorphism. illustrates 1 way first-order system pa weaker second-order peano axioms.

when interpreted proof within first-order set theory, such zfc, dedekind s categoricity proof pa shows each model of set theory has unique model of peano axioms, isomorphism, embeds initial segment of other models of pa contained within model of set theory. in standard model of set theory, smallest model of pa standard model of pa; however, in nonstandard model of set theory, may nonstandard model of pa. situation cannot avoided first-order formalization of set theory.

it natural ask whether countable nonstandard model can explicitly constructed. answer affirmative skolem in 1933 provided explicit construction of such nonstandard model. on other hand, tennenbaum s theorem, proved in 1959, shows there no countable nonstandard model of pa in either addition or multiplication operation computable. result shows difficult explicit in describing addition , multiplication operations of countable nonstandard model of pa. however, there 1 possible order type of countable nonstandard model. letting ω order type of natural numbers, ζ order type of integers, , η order type of rationals, order type of countable nonstandard model of pa ω + ζ·η, can visualized copy of natural numbers followed dense linear ordering of copies of integers.

set-theoretic models

the peano axioms can derived set theoretic constructions of natural numbers , axioms of set theory such zf. standard construction of naturals, due john von neumann, starts definition of 0 empty set, ∅, , operator s on sets defined as:

s(a) = ∪ { }.

the set of natural numbers n defined intersection of sets closed under s contain empty set. each natural number equal (as set) set of natural numbers less it:

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{\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}

and on. set n 0 , successor function s : n → n satisfies peano axioms.

peano arithmetic equiconsistent several weak systems of set theory. 1 such system zfc axiom of infinity replaced negation. such system consists of general set theory (extensionality, existence of empty set, , axiom of adjunction), augmented axiom schema stating property holds empty set , holds of adjunction whenever holds of adjunct must hold sets.

interpretation in category theory

the peano axioms can understood using category theory. let c category terminal object 1c, , define category of pointed unary systems, us1(c) follows:

the objects of us1(c) triples (x, 0x, sx) x object of c, , 0x : 1c → x , sx : x → x c-morphisms.
a morphism φ : (x, 0x, sx) → (y, 0y, sy) c-morphism φ : x → y φ 0x = 0y , φ sx = sy φ.

then c said satisfy dedekind–peano axioms if us1(c) has initial object; initial object known natural number object in c. if (n, 0, s) initial object, , (x, 0x, sx) other object, unique map u : (n, 0, s) → (x, 0x, sx) such that

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{\displaystyle {\begin{aligned}u0&=0_{x},\\u(sx)&=s_{x}(ux).\end{aligned}}}

this precisely recursive definition of 0x , sx.