Note: real analysis is not on the MIRI reading list (although I think it should be).
Foreword
As a young boy, mathematics captivated me.
In elementary school, I'd happily while away entire weekends working through the next grade's math book. I was impatient.
In middle school, I'd lazily estimateangles of incidence that would result if I shot lasers from my eyes, tracing their trajectories within the classroom and out down the hallway. I was restless.
In high school, I'd daydream about what would happen to integrals as I twisted functions in my mind. I was curious.
And now, I get to see how it's all put together. Imagine being fascinated by some thing, continually glimpsing beautiful new facets and sampling exotic flavors, yet being resigned to not truly pursuing this passion. After all, I chose to earn a computer science degree.
Wait.
Analysis I
As in Linear Algebra Done Right, I completed every single exercise in the book - this time, without looking up any solutions (although I did occasionally ask questions on Discord). Instead, I came back to problems if I couldn't solve them after half an hour of effort.
In which the Peano axioms are introduced, allowing us to define addition and multiplication on the natural numbers {0,1,2,…}.
3: Set Theory
In which functions and Cartesian products are defined, among other concepts.
Recursive Nesting
How can you apply the axiom of foundation if sets are nested in each other? That is, how can the axiom of foundation "reach into" sets like A={B,…} and B={A,…}?
Show that if A and B are two sets, then either A∉B or B∉A (or both).
Note: real analysis is not on the MIRI reading list (although I think it should be).
Foreword
As a young boy, mathematics captivated me.
In elementary school, I'd happily while away entire weekends working through the next grade's math book. I was impatient.
In middle school, I'd lazily estimate angles of incidence that would result if I shot lasers from my eyes, tracing their trajectories within the classroom and out down the hallway. I was restless.
In high school, I'd daydream about what would happen to integrals as I twisted functions in my mind. I was curious.
And now, I get to see how it's all put together. Imagine being fascinated by some thing, continually glimpsing beautiful new facets and sampling exotic flavors, yet being resigned to not truly pursuing this passion. After all, I chose to earn a computer science degree.
Wait.
Analysis I
As in Linear Algebra Done Right, I completed every single exercise in the book - this time, without looking up any solutions (although I did occasionally ask questions on Discord). Instead, I came back to problems if I couldn't solve them after half an hour of effort.
A sampling of my proofs can be found here.
1: Introduction
2: The Natural Numbers
In which the Peano axioms are introduced, allowing us to define addition and multiplication on the natural numbers {0,1,2,…}.
3: Set Theory
In which functions and Cartesian products are defined, among other concepts.
Recursive Nesting
How can you apply the axiom of foundation if sets are nested in each other? That is, how can the axiom of foundation "reach into" sets like A={B,…} and B={A,…}?
Show that if A and B are two sets, then either A∉B or B∉A (or both).
Proof. Suppose A∈B and