Some Theorems Derivable from Peano's Axioms

The following is quoted from Edmund Landau, Foundations of Analysis, Chelsea, 1951, pp. 1-18.

Theorem 1:

If then .

Theorem 2:

.

Theorem 3:

If there exists one (and hence, by Axiom 4, exactly one) u such that x=u'.

Theorem 4,

and at the same time Definition 1: To every pair of numbers x,y, we may assign in exactly one way a natural number, called x+y (+ to be read ``plus''), such that

1.

x+1=x' for every x.

2.

x+y'=(x+y)' for every x and every y.

x+y is called the sum of x and y, or the number obtained by addition of y to x.

Theorem 5 (Associative Law of Addition):

(x+y)+z=x+(y+z).

Theorem 6 (Commutative Law of Addition):

x+y=y+x.

Theorem 7:

.

Theorem 8:

If then .

Theorem 9:

For given x and y, exactly one of the following must be the case:

1.

x=y.

2.

There exists a u (exactly one, by Theorem 8) such that x=y+u.

3.

There exists a v (exactly one, by Theorem 8) such that y=x+v.

Definition 2:

If x=y+u then x>y. (> to be read ``is greater than.'')

Definition 3:

If y=x+v then x<y. (< to be read ``is less than.'')

Theorem 10:

For any given x,y, we have exactly one of the cases x=y, x>y, x<y.

Theorem 11:

If x>y then y<x.

Theorem 12:

If x<y then y>x.

Definition 4:

means x>y or x=y. ( to be read ``is greater than or equal to.'')

Definition 5:

means x<y or x=y. ( to be read ``is less than or equal to.'')

Theorem 13:

If then .

Theorem 14:

If then .

Theorem 15 (Transitivity of Ordering:)

If x<y, y<z, then x<z.

Theorem 16:

If , y<z or x<y, , then x<z.

Theorem 17:

If , , then .

Theorem 18:

x+y>x.

Theorem 19:

If x>y, or x=y, or x<y, then x+z>y+z, or x+z=y+z, or x+z<y+z, respectively.

Theorem 20:

If x+z>y+z, or x+z=y+z, or x+z<y+z, then x>y, or x=y, or x<y, respectively.

Theorem 21:

If x>y, z>u, then x+z>y+u.

Theorem 22:

If , z>u or x>y, , then x+z>y+u.

Theorem 23:

If , , then .

Theorem 24:

.

Theorem 25:

If y>x then .

Theorem 26:

If y<x+1 then .

Theorem 27:

In every non-empty set of natural numbers there is a least one (i.e., one which is less than any other number of the set).

Theorem 28

and at the same time Definition 6: To every pair of numbers x,y, we may assign in exactly one way a natural number, called ( to be read ``times''; however, the dot is usually omitted), such that

1.

for every x,

2.

for every x and every y.

is called the product of x and y, or the number obtained from multiplication of x by y.

Theorem 29 (Commutative Law of Multiplication):

xy=yx.

Theorem 30 (Distributive Law):

x(y+z)=xy+xz.

Theorem 31 (Associative Law of Multiplication):

(xy)z=x(yz).

Theorem 32:

If x>y, or x=y, or x<y, then xz>yz, xz=yz, or xz<yz, respectively.

Theorem 33:

If xz>yz, or xz=yz, or xz<yz, then x>y, or x=y, or x<y, respectively.

Theorem 34:

If x>y, z>u, then xz>yu.

Theorem 35:

If , z>u or x>y, , then xz>yu.

Theorem 36:

If , , then .