Theory with miltiplication and ordering as a theory of N
by
Kamila Bendova
Charles University, Philosophical Faculty, Dep.of Logic, Celetna 20, 110 00 Prague, Czech Republic

Let T be a theory with equality constant 1 and one operation and one basic binary predicate: multiplication · and ordering  < . Axioms

1. x ·y = y ·x                    commutativity
2. x ·(y ·z) = (x ·y) ·z        associativity
3. 1 ·x = x                         unary element
4. the usual axioms stating that  < is a linear discrete order with the smallest element 1; there is no largest element. S(x) is a successor of x.
5. S(1) = 2  and  2 ·S(x) = SS(2 ·x) = S²(2 ·x)
6. x < y ===> x ·z < y ·z
7. [x < y & (x ·z < y ·w   \/  x ·z = y ·w)] ===> x ·S(y) < y ·S(w)
8. In a special case:


S(x ·SS(x)) = S(x) ·S(x)


9. [x < y & (x ·z < y ·w   \/  x ·z = y ·w )] ===> y ·P(z) < x ·P(w)

Definition.
Let us define n = SSS...S(1) = S^n-1(1)
and (by Julia Robinson) x+y=z iff (xz + 1) ·(yz+1) = [z² ·(xy +1)] + 1

Lemma.
m + 1 = S(m)
m + S(n) = S(m+n)
m ·S(n) = n ·m + m.

Theorem.
The theory T is \Sigma₁ complete.

References.

Bendová K.: On ordering and multiplication of natural numbers, Archive for Mathematical Logic (2001) 40: 19-23

Date received: March 21, 2002