1 - What Gödel's Theorems say

Summary

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.