Defining Universal Sets

We've already defined whole numbers casually by {0, 1, 2, ...} and formally using Peano's Axioms. Let's analyze the Peano Axioms and understand why they work.

Set theory isn't something that came naturally in mathematics. The general idea comes naturally to us, of course, but it's so difficult to actually formalize that it wasn't done until Georg Cantor's work in the 19th century. Even then, there were problems. Bertrand Russell found a paradox in "naive set theory" in 1901, and even still inconsistencies may exist. Today, much work has been done in axiomatic set theory. I'm not going to get in to these axioms, but I will talk about how I like to define sets.

Most sets that we think of, such as {1, 4, 5} are subsets of some universe. In this case, the universe can be whole numbers, or it can be integers or reals. Defining these subsets isn't hard, it's defining the universal sets that proves tricky. Once you have the universal set, subsets can be defined using some rule that separates values in the subset from values outside the subset.

So how do we define these universal sets? My ideas on this have been shaped by ideas in computer science, which looks at data structures and functions. The way I tend to look at it, there are 3 ingredients that define U.

1) Constructors. These are functions that accept as input members of various other well-defined sets and possibly U as well, and use these inputs to construct a value in U.

2) Induction Rules. Suppose that X is a subset of U. We need a way to prove or disprove that X contains all of U.

3) Equality Rules. We need to know which members of the set are equal to one another. If we do not know this, we cannot know the precise structure of the set. How do we know that our definition of equality is legit? It needs to be defined for all values on U, which we can prove using our induction rules, and it must also satisfy the 3 properties of equivalence relations. That is,
Reflection: a = a
Symmetry: a = b implies b = a
Transition: if a = b and b = c then a = c.

Once these 3 rules are defined, we have a functioning universal set.