#### anonimnystefy wrote:

It cannot be prpved. It is an axiom of natural numbers and of higher number sets, too.

It depends on how you are defining your sets of numbers. If you are defining them by axioms, then there is nothing to prove since the axioms will include commutativity and associativity of addition. But if you are constructing them from smaller sets of numbers, then commutativity and associativity need to be proved.

For example, here is briefly how

is constructed from

. Define a relation

on

by

iff

. (Hint: Think of

as the "difference"

.) Then

is an equivalence relation and

is defined as the set of all equivalence classes under

. Let the equivalence class containing

be denoted

. Then addition in

is defined as

After checking that the operation is well defined, one can proceed to verify that

is commutative and associative.

Moreover, zero

in

is the equivalence class

and multiplication in

is defined as

.

Similarly

can be constructed from

as equivalence classes of the equivalence relation

on

defined by

iff

. Letting the equivalence class containing

be denoted

addition in

is definied as