Ordinal analysis

Ordinal analysis is like building with blocks. Imagine you have lots of blocks of different colors and shapes, and you want to build a really tall tower. You can put the blocks on top of each other and make the tower as tall as you want.

Now imagine instead of blocks, you have statements in a language, and instead of building a tower, you're trying to prove that a particular statement is true. To do this, you have to start with some basic statements (like blocks) that are already true, and then use certain rules (like how to stack the blocks) to build up to the statement you want to prove.

Ordinal analysis is a method for figuring out exactly which basic statements you need to start with, and exactly which rules you need to use, in order to prove a particular statement. It's kind of like having a recipe that tells you exactly which ingredients and steps you need to follow to make a cake.

This method is useful for studying really complicated mathematical theories, because it allows you to break down big, complicated problems into smaller, more manageable ones. And by doing this, mathematicians can figure out new things about how different mathematical systems work, and discover new connections between seemingly unrelated topics.