A disruptive view of big number arithmetic | Data structures in Mathematics Math Foundations 173

What do really big numbers look like? What does thinking about such objects tell us about arithmetic and mathematics more generally? It certainly gives us a dose of reality with regard to the current overblown claims that modern mathematics “understands the infinite“. In this video we directly explore some pretty big numbers, like a googol, and a googolplex, and even much bigger ones. We will introduce some disturbing consequences of our inability to do arithmetic with them. This is similar to what happens in “real number arithmetic“, when empty phrases and claims become substitutes for actual computations and examples. We also see that induction as a proof technique breaks down, and that even the nature of a natural number is ultimately called into question. So not for the faint-hearted. Video Content: 00:00 Intro to really big arithmetic! 2:18 Extremely big numbers 6:24 Using towers of exponents 9:36 Towers of towers of exponents 17:45 Consequences