MAT 371 Advanced Calculus
Starting with the Peano axioms
(Adapted from http://149.150.2.6/projects/reals/logic/defs/peano.html)
Peano axioms:
•1 is a natural number
•For every natural number xthere exists a natural number x0called the successor of x.
•16=x0for every natural number x.
•If x, y are natural numbers and x0=y0then x=y
•If Pis a property such that:
–1 has the property P, and
–whenever xis a natural number and xhas property Pthen x0has property P.
then all natural numbers have the property P.
Adapted from http://sisenis.com.latnet.lv/˜podnieks/gt3.html
The modern version of Peano axioms can be put as follows. Let variables x,y, range over
natural numbers, and let 0 denote the number ”zero”, x0- denote the operation x+ 1, and let
the variable Srange over sets of natural numbers. Then the following statements are called
Peano axioms:
P1) ∀x, (0 6=x0)
P2) ∀x∀y, ((x6=y)−→ (x06=y0)
P3) (0 ∈S)∧ ∀x(x∈S−→ x0∈S)) −→ ∀x(xinS)
One can prove easily that any two “systems” IN and ˜
IN satisfying the axioms (P1)-(P3) are
isomorphic . . . . One of such “systems” can be constructed in Zermelo-Fr¨ankel ZF set theory:
Define 0 as the empty set, and x0=x∪ {x}, and let variables range over the set ω. . . .
From here, define: x+ 1 = x0, and “inductively” x+y0= (x+y)0,
then prove associativity and commutativity of (IN,+).
Define x·1 = xand “inductively” x·y0= (x·y) + y,
and prove associativity and commutativity of (IN,+), establish the distributive laws.
Proceed along either of the following ways:
In order to be able to solve all equations m+x=n(with m, n ∈IN) define the set ZZ of
integers as the set of equivalence classes (m, n)∈IN ×IN under the relation (m, n)∼(r, s) iff
m+s=n+r. Define subtraction (of equivalence classes) . . .
In order to be able to solve all equations q∗x=p(with p, q ∈IN, q6= 0) define the set Q+
of positive rational numbers as the set of equivalence classes (p, q )∈IN ×(IN \ {0}) under the
relation (p, q)∼(r, s) iff p·s=q·t. Define division (of equivalence classes) . . .
In either case check that the usual rules of arithmetic still hold, especially check that associativ-
ity and commutativity carry over from IN. Then carry out the other step as well to eventually
obtain the set of rational numbers Q.