Arithmetic Peano axioms

1 arithmetic

1.1 addition
1.2 multiplication
1.3 inequalities

arithmetic

the peano axioms can augmented operations of addition , multiplication , usual total (linear) ordering on n. respective functions , relations constructed in set theory or second-order logic, , can shown unique using peano axioms.

addition

addition function maps 2 natural numbers (two elements of n) one. defined recursively as:

a
+
0



=
a
,




(1)






a
+
s
(
b
)



=
s
(
a
+
b
)
.




(2)








{\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+s(b)&=s(a+b).&{\textrm {(2)}}\end{aligned}}}

for example:

a
+
1



=
a
+
s
(
0
)



definition







=
s
(
a
+
0
)



using (2)







=
s
(
a
)
,



using (1)








a
+
2



=
a
+
s
(
1
)



definition







=
s
(
a
+
1
)



using (2)







=
s
(
s
(
a
)
)



using 

a
+
1
=
s
(
a
)







a
+
3



=
a
+
s
(
2
)



definition







=
s
(
a
+
2
)



using (2)







=
s
(
s
(
s
(
a
)
)
)



using 

a
+
2
=
s
(
s
(
a
)
)





etc.








{\displaystyle {\begin{aligned}a+1&=a+s(0)&{\mbox{by definition}}\\&=s(a+0)&{\mbox{using (2)}}\\&=s(a),&{\mbox{using (1)}}\\\\a+2&=a+s(1)&{\mbox{by definition}}\\&=s(a+1)&{\mbox{using (2)}}\\&=s(s(a))&{\mbox{using }}a+1=s(a)\\\\a+3&=a+s(2)&{\mbox{by definition}}\\&=s(a+2)&{\mbox{using (2)}}\\&=s(s(s(a)))&{\mbox{using }}a+2=s(s(a))\\{\text{etc.}}&\\\end{aligned}}}

the structure (n, +) commutative monoid identity element 0. (n, +) cancellative magma, , embeddable in group. smallest group embedding n integers.

multiplication

similarly, multiplication function mapping 2 natural numbers one. given addition, defined recursively as:

a
⋅
0



=
0
,




a
⋅
s
(
b
)



=
a
+
(
a
⋅
b
)
.






{\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot s(b)&=a+(a\cdot b).\end{aligned}}}

it easy see s(0) (or 1 , in familiar language of decimal representation) multiplicative right identity:

a · s(0) = + (a · 0) = + 0 = a

to show s(0) multiplicative left identity requires induction axiom due way multiplication defined:

s(0) left identity of 0: s(0) · 0 = 0.
if s(0) left identity of (that s(0) · = a), s(0) left identity of s(a): s(0) · s(a) = s(0) + s(0) · = s(0) + = + s(0) = s(a + 0) = s(a).

therefore induction axiom s(0) multiplicative left identity of natural numbers. moreover, can shown multiplication distributes on addition:

a · (b + c) = (a · b) + (a · c).

thus, (n, +, 0, ·, s(0)) commutative semiring.

inequalities

the usual total order relation ≤ on natural numbers can defined follows, assuming 0 natural number:

for a, b ∈ n, ≤ b if , if there exists c ∈ n such + c = b.

this relation stable under addition , multiplication:



a
,
b
,
c
∈

n



{\displaystyle a,b,c\in \mathbf {n} }

, if ≤ b, then:

a + c ≤ b + c, and
a · c ≤ b · c.

thus, structure (n, +, ·, 1, 0, ≤) ordered semiring; because there no natural number between 0 , 1, discrete ordered semiring.

the axiom of induction stated in following form uses stronger hypothesis, making use of order relation ≤ :

for predicate φ, if

φ(0) true, and
for every n, k ∈ n, if k ≤ n implies φ(k) true, φ(s(n)) true,


then every n ∈ n, φ(n) true.

this form of induction axiom, called strong induction, consequence of standard formulation, better suited reasoning ≤ order. example, show naturals well-ordered—every nonempty subset of n has least element—one can reason follows. let nonempty x ⊆ n given , assume x has no least element.

because 0 least element of n, must 0 ∉ x.
for n ∈ n, suppose every k ≤ n, k ∉ x. s(n) ∉ x, otherwise least element of x.

thus, strong induction principle, every n ∈ n, n ∉ x. thus, x ∩ n = ∅, contradicts x being nonempty subset of n. x has least element.