Goodstein's theorem

Goodstein's theorem is like a big puzzle game with numbers. Let’s imagine that we have numbers written down on a piece of paper, like 2, 3, 4, 5, and 6.

Now, we stack these numbers on top of each other, with the bigger numbers on top and the smaller numbers at the bottom. We call this stack B.

Next, let’s do some math. Take the number at the bottom of the stack (in this case, 2), and subtract 1 from it. So now we have 1. We replace the number at the bottom of the stack with this new number.

But we’re not done yet! Now we do the same thing with the next number in the stack (in this case, 3). Subtract 1 from it, and you get 2. Replace 3 with 2 in the stack.

Continue doing this with each number in the stack, and you’ll get a new stack of numbers (let’s call this one B1). We’ll do the same process again on B1 to get B2, and so on.

What Goodstein's theorem says is that no matter what numbers we start with, we will eventually get to a point where we have a stack of zeros. In other words, no matter how big our starting numbers are, if we keep doing these steps over and over again, we will eventually end up with a stack of zeros.

This may seem strange or hard to believe, but mathematicians have proven that it’s true! And it’s a very important discovery for understanding how numbers work.