Blogs: Unassigned Entry #2: What's in a Number?

   Numbers. We all use them practically every day. We basically take them for granted, but what exactly is a number? How can we know that our conception of numbers even make sense? The first question is a major topic in the philosophy of mathematics, and I will not answer is, at least not today. The second question however, is today's topic.

   On the sensibility of numbers, of mathematics, rigor has been the strategy for centuries. The thinking is if you have a short set of rules, and prove everything from these rules, everything will make sense. In 1931, Kurt Gödel proved that no system can ever be know to be logical, they are either illogical or not yet proven to be illogical. Despite this, the rules-based approach is the best thing we have.

   So what are the basic rules, or axioms, of numbers? There are many different numbers; natural, integer, ration, real, complex, quarterion, hyperreal, surreal, the list goes on and on. Limiting the discussion to just natural numbers (0, 1, 2, 3, ...)

1. Each natural number is a list, or set. 0 is a set with nothing in it; {}.
2. 0 is a natural number.
3. Each natural number is the set of all smaller numbers. i.e. 1 = {0}, 3 = {0, 1, 2}.
4. Each natural number has a successor number, the number that comes next, represented s(x). e.i. s(0) = 1, s(12345) = 12346.
5. No natural number's successor is 0; 0 comes after no number.
6. If a natural number succeeds two numbers, they are the same number, i.e. If s(x) = 17 and s(y) = 17, then x = y = 16.
7. All natural numbers are a result of finite applications of succession to 0. Thus, s(s(s(s(0)))) is a natural number (namely 4), but s(s(...)) is not because of infinite succession. {0, 1, 4} is not a number because it cannot be formed by succession; you cannot add 4 before adding 3.

These rules together define the natural numbers; all other numbers can be defined in terms of the naturals and there are other rules for addition, multiplication, etc.