Filed under: Algebra, Math | Tags: abelian groups, algebra, arithmetic, field theory, group theory, Math, ring theory
In which I sort of breeze through a couple of really awesome and really important concepts. Last time, we classified abelian groups — now we’ll see what happens if we require additional structure on the groups. In particular, I’m going to construct and similarly to how the Peano axioms constructed .
Filed under: Math, Set Theory | Tags: arithmetic, Math, ordinals, set theory
Freeing yourself of the bounds of Peano arithmetic, you come upon a number of different routes to take. You want the addition and multiplication operations you defined to make sense in a larger scope, but what this scope is varies depending on your needs. The standard thing to do is to start “closing” the ordered semiring in a number of different ways: group closure by adding negatives and reciprocals, metric closure by adding limits, or algebraic closure by adding solutions to polynomials. But instead, we could just model the elements of as the finite ordinals and use transfinite recursion to extend the operations to operations on ordinals. Transfinite recursion is like induction on cagefighting elephant steroids.
Once you’ve done this, you lose some of your semiring properties, but you retain the order and you win the exciting new property of it being closed under limits. This is ordinal arithmetic. Building it will let us understand the structure of the ordinals more rigorously than we had before: there was an ordinal we called , but what does this actually mean?
Let’s do something fun!
The Peano axioms for arithmetic are something like the Zermelo-Fraenkel axioms for sets. They describe what we intuitively want to be true about the natural numbers. Using ZFC, we were able to prove the existence of an inductive set , and it’s actually pretty easy to prove that the axioms below hold for that set. But the idea of the Peano axioms is that any structure satisfying them is “just as good” a model for the natural numbers. That is, it’s a necessary fact about that every number has a successor, but it’s not a necessary fact that the successor of is the set . It’s not even necessary that be a set at all.
And in fact, there are other models. We could take a category-theory approach and look at as the initial object in a category of things satisfying the axioms. We could treat numbers as just numbers and not suppose any internal structure. It doesn’t really matter, and perhaps this is the reason why we axiomatize stuff, so that we know what matters and what doesn’t.
The following is meant to be a bit of an exploration, mirroring Peano’s own exposition, into the properties of the natural numbers. Many of them you probably saw in school, but now you can actually prove them from a very small set of definitions and axioms. If nothing else, it’s a nice exercise in thinking logically.