In other words, what’s the characteristics of numbers?
Numbers and arithmetic are made for each other – a number is a number only if arithmetical properties operate on it. Peano arithmetic may be used to create a definition for numbers to test if a number is really a number. Peano arithmetic is a set of ‘rules’/self-evident truths (axioms) that defines how natural numbers work – rules that can be used as the fundamental basis for seeking a formal definition of numbers.
For instance, some of the important properties of numbers are –
- For every pair of (natural) numbers a and b, the sum a + b is well defined
- For every (natural) number, a, 0 + a = a holds true
- For every pair of (natural) numbers a and b, a + b = b + a
- For every (natural) number, there is exactly one natural number that is its successor; and every natural number except zero is the successor of exactly one natural number.
- Distinct (natural) numbers have distinct successors.
- Similarly, for every natural number, there is one unique predecessor too, except for 1 (if we ignore 0).
At this point the only operations we have are successor and predecessor. But any self-respecting theory of arithmetic also needs addition and multiplication. Our strategy for developing these operations is simple: we define addition in terms of iterated successor, and multiplication in terms of iterated addition.
Continuing on, we will define exponentiation in terms of iterated multiplication. These definitions all rely on the general concept of iteration, so in order to reach our goal of basic arithmetic, we need to take a side trip through iteration. Informally, we can think of iteration in terms of repeating an action or processes. In these notes we think of operations, actions, processes, and such in terms of functions. So iteration will mean repeatedly applying a function.
We define addition in terms of iteration of successor. Informally, you get m + n by starting with m and taking the successor n times. This idea motivates the formal definition.
Thus, we have the definition of addition in terms of iteration of the successor function, multiplication can be explained of in terms of iteration of addition, and how exponentiation can be explained in terms of iteration of multiplication.