The Structure Theorem over PID’s

In this post I'll describe the structure theorem over PID's which generalizes the following results: Finite dimensional vector fields over $latex {k}&fg=000000$ are all of the form $latex {k^{\oplus n}}&fg=000000$, The classification theorem for finitely generated abelian groups, The Frobenius normal form of a matrix, The Jordan decomposition of a matrix. 1. Some ring theory… Continue reading The Structure Theorem over PID’s

Models of ZFC

Model theory is really meta, so you will have to pay attention here. Roughly, a ``model of $latex {\mathsf{ZFC}}&fg=000000$'' is a set with a binary relation that satisfies the $latex {\mathsf{ZFC}}&fg=000000$ axioms, just as a group is a set with a binary operation that satisfies the group axioms. Unfortunately, unlike with groups, it is very… Continue reading Models of ZFC


(Standard post on cardinals, as a prerequisite for forthcoming theory model post.) An ordinal measures a total ordering. However, it does not do a fantastic job at measuring size. For example, there is a bijection between the elements of $latex {\omega}&fg=000000$ and $latex {\omega+1}&fg=000000$: $latex \displaystyle \begin{array}{rccccccc} \omega+1 = & \{ & \omega & 0… Continue reading Cardinals