I am reading the lecture notes and have a question. I am trying to understand the beginning of Section 3 on page 2. 

Problem: Given an input stream $\sigma$, compute (or approximate) its length $m$.

Naive solution: $O(\log m)$ bits, exact solution.

I don't understand why it is not $O(m)$ bits but $O(\log m)$ bits. Any help would be greatly appreciated. 

There is no formal definition of "safe programming language"; it's an informal notion.  Rather, languages that claim to provide safety usually provide a precise formal statement of what kind of safety is being claimed/guaranteed/provided.  For instance, the language might provide type safety, memory safety, or some other similar guarantee.