Our first "definition" of ℕ will characterize it in a set theoretical framework. This way of starting to formalize ℕ now may look circular, as we already used natural numbers as arities of operation symbols of algebras, of which arithmetic is a particular case. But this case only uses operation symbols with arity 0, 1 or 2, for which previous definitions might as well be specially rewritten without any general reference to numbers.

The interpretation of *S* there is called the *successor*, understood as adding one
unit (*Sn*=*n*+1).

∀n∈ℕ, Sn ≠ 0 |
(H0), i.e. 0 ∉ Im S |

∀n,p∈ℕ, Sn =
Sp ⇒ n = p |
(Inj), i.e. S is injective |

∀A⊂ℕ, (0∈A ∧ ∀n∈A,Sn∈A)
⇒ A=ℕ |
(Ind) : induction axiom (ℕ is a minimal (0,S)-algebra). |

We can define 1=*S*0, 2=*SS*0...

In particular, a

*u*_{0}=*a*

∀*n*∈ℕ, *u*_{Sn} = *f*(*u _{n}*).

As an element of a ground term {0,

f^{ 0}(a) = af^{ 1}(a) = f(a)f^{ 2}(a) = f(f(a)) |

∀

More generally, for any functions

*n* + 0 = *n*

∀*p*∈ℕ, *n*+*S*(*p*) = *S*(*n*+*p*).

∀*n,p*∈ℕ , *f ^{ n+p}* =

It is also commutative as it is generated by 1 which commutes with itself.

Multiplication in ℕ can be defined as *x*⋅*y* =
(*S ^{x}*)

∀x∈ℕ,x⋅0 = 0

∀x,y∈ℕ,x⋅(Sy) = (x⋅y)+x

More generally, for any *a*∈*E* and *f*∈ *E ^{E}*,
we have

∀*f*,*g*∈
*E ^{E}*,

Commutativity was just here to show that composing

(*f*^{ Sn})^{-1} =
(*f*^{n})^{-1}০*f*.

- the set ℤ = ℕ ∪ -ℕ, where natural numbers (in ℕ) are
called
*positive*, and-ℕ= {-

is a copy of ℕ called the set of negative integers (where -*n*|*n*∈ℕ}*n*is the*opposite*of*n*), only intersecting ℕ at -0 = 0. - The interpretation of
*S*in ℕ is extended to a permutation of ℤ by ∀*n*∈ℕ,*S*(-*Sn*)= -*n*, thus letting Gr*S*_{ℤ}be the union of Gr*S*_{ℕ}with its transposed copy in -ℕ.

Proof: the {0,*S*}-morphism condition requires on ℕ the same *n* ↦ *f ^{ n}* as above;
on -ℕ, it recursively defines

*f*^{ -0}= Id_{E}=*f*^{ 0}- ∀
*n*∈ℕ,*f*^{ -n}=*f*০*f*, equivalently^{ -Sn}*f*^{ -1}০*f*^{ -n}=*f*^{ -Sn}

This monoid is commutative because it is generated by {-1, 1}, which commute: (-1)+1=0=1+(-1).

It is a group: (-

For any algebraic language

The version we saw was formalized by giving the term in the recursive definition as an

∀(*s*,*x*)∈*L*⋆*E*,
*u*(*s _{E}*(

*u*(*s _{E}*(

The first component (φ

Id

It is then possible to conclude by re-using the previous result of existence of interpretations:

IfBut one can do without it, based on the following property of thisEis a closed termL-algebra then ∃!f∈ Mor(E,E×F), which is of the form Id_{E}×ubecause π০f∈ Mor(E,E) ∴ π০f= Id_{E}.

∀*u*∈*F*^{E}, Id_{E}×*u*
∈ Mor_{L}(*E*, *E*×*F*) ⇔ Gr *u*
∈ Sub_{L}(*E*×*F*)

- ⇒ is a case of image of an algebra by a
morphism, Gr
*u*= Im (Id_{E}×*u*). -
For the converse, the inverse of the bijective morphism π
_{|Gr u}∈ Mor_{L}(Gr*u*,*E*) is a morphism Id_{E}×*u*∈ Mor_{L}(*E*, Gr*u*) ⊂ Mor_{L}(*E*,*E*×*F*).

Now if

Any other subalgebra of

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3.1. Morphisms of relational systems and concrete categories4. Model Theory

3.2. Algebras

3.3. Special morphisms

3.4. Monoids

3.5. Actions of monoids

3.6. Categories

3.7. Algebraic terms and term algebras

3.8.Integers and recursion

3.9. Arithmetic with addition