To Big Finities and Beyond: The Climb Above the TREEline is a short book I'm writing about big numbers, things like a googol; a googolplex; a googolplex to the power of a googolplex; a googolplex to the power a googolplex to the power of a googolplex and so on a googolplex times; and so on. It might seem that the "and so on" here gestures at some pretty big numbers, but that doesn't even scratch the surface of the sort of big numbers this book is about.
You might have heard of the number TREE(3), which is the solution to a certain problem in graph theory that is quite easy to state, but which is a very very big number. How big is it? Normal answers to this question simply don't do it justice. In this book, I try to.
Now, you might not think that there is much of philosophical interest in big numbers like this, but, in fact, the quest to define bigger and bigger numbers leads one into some of the most philosophically interesting branches of math that there are, such as ordinal analysis and its applications in proof theory, reverse mathematics, computability theory, and the theory of formal languages. Beyond just being about big numbers, this book is also a philosophical introduction to those fields of math and logic through the quest to define bigger and bigger numbers.
(And yes, I'm using AI to illustrate it. I would do it myself, but this book is already eating up too much of my time.)
You might have heard of the number TREE(3), which is the solution to a certain problem in graph theory that is quite easy to state, but which is a very very big number. How big is it? Normal answers to this question simply don't do it justice. In this book, I try to.
Now, you might not think that there is much of philosophical interest in big numbers like this, but, in fact, the quest to define bigger and bigger numbers leads one into some of the most philosophically interesting branches of math that there are, such as ordinal analysis and its applications in proof theory, reverse mathematics, computability theory, and the theory of formal languages. Beyond just being about big numbers, this book is also a philosophical introduction to those fields of math and logic through the quest to define bigger and bigger numbers.
(And yes, I'm using AI to illustrate it. I would do it myself, but this book is already eating up too much of my time.)