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 it 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 integers.

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,p*∈ℕ , *f*^{ n+p} = *f*^{
p}০*f*^{ n}.

Addition is associative: (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

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*)

For the converse, the inverse of the bijective morphism π

This reduces the issue to the search of subalgebras of

Now if

Any other subalgebra of

*Commutative*when ∀*x*,*y*∈*E*,*x*#*y*=*y*#*x*.*Associative*when ∀*x*,*y*,*z*∈*E*,*x*#(*y*#*z*) = (*x#y*)#*z*, so that we can write*x*#*y*#*z*to mean either of these terms.

This is a Galois connection: ∀*A*,*B*⊂*E*, *B*⊂*C*(*A*)
⇔ *A*⊂*C*(*B*).

*C*(*E*)= *E* expresses the commutativity of #.

*C*(*A*) ∈ Sub_{#}*F*- If
*A*⊂*C*(*A*) then # is commutative in 〈*A*〉_{#}

- ∀
*x,y*∈*C*(*A*), (∀*z*∈*A*, (*x*#*y*)#*z*=*x*#*z*#*y*=*z*#(*x#y*)) ∴*x*#*y*∈*C*(*A*) *A*⊂*C*(*A*)∈ Sub_{#}*F*⇒〈*A*〉_{#}⊂*C*(*A*) ⇒*A*⊂*C*(〈*A*〉_{#})∈ Sub_{#}*F*⇒ 〈*A*〉_{#}⊂*C*(〈*A*〉_{#}).

More texts on algebra

Back to homepage : 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. Special morphisms

3.3. Algebras

3.4. Algebraic terms and term algebras

3.5.Integers and recursion

3.6. Arithmetic with addition

4.1. Finiteness and countability

4.2. The Completeness Theorem

4.3. Infinity and the axiom of choice

4.4. Non-standard models of Arithmetic

4.5. How theories develop

4.6. The Incompleteness Theorem