Basic arithmetic

It seems to be child's play to grasp the fundamental notions involved in the arithmetic of addition and multiplication. Starting from zero, there is a sequence of ‘counting’ numbers, each having just one immediate successor. This sequence of numbers – officially, the natural numbers – continues without end, never circling back on itself; and there are no ‘stray’ numbers, lurking outside this sequence. Adding n to m is the operation of starting from m in the number sequence and moving n places along. Multiplying m by n is the operation of (starting from zero and) repeatedly adding m, n times. And that's about it.

Once these fundamental notions are in place, we can readily define many more arithmetical notions in terms of them. Thus, for any natural numbers m and n, m < n iff there is a number k ≠ 0 such that m + k = n. m is a factor of n iff 0 < m and there is some number k such that 0 < k and m × k = n. m is even iff it has 2 as a factor. m is prime iff 1 < m and m's only factors are 1 and itself. And so on.

Using our basic and/or defined concepts, we can then make various general claims about the arithmetic of addition and multiplication. There are familiar truths like ‘addition is commutative’, i.e. for any numbers m and n, we have m + n = n + m.