Cantor's first uncountability proof

Imagine you have a bunch of toys in a toy box. You can count all of them by taking one out at a time and saying the number of toys you have. Now imagine you have an infinite number of toys. You might think you can still count them by taking one out at a time, but actually, there are some things that you can't count by just taking them out one by one.

For example, let's say you have an infinite number of numbers between 0 and 1. You might think that you can count them by taking one out at a time and saying the number you have, but actually, you can't. This is because there are an infinite number of these numbers, and they are not all whole numbers. If you try to count them, you will never be able to finish because there will always be more numbers you haven't counted yet.

This is what Cantor's first uncountability proof is about. Cantor showed that there are some things that are too big to be counted by just taking them out one by one. He did this by showing that there are more real numbers between 0 and 1 than there are whole numbers (which are countable).

Cantor's proof is complex, but you can think of it like this: imagine you have a list of all the real numbers between 0 and 1. Cantor showed that you can always find a real number that is not on that list, no matter how long or how complete the list is. This means that there are always more real numbers than you can count on any list, and so you can never count them all.

So, just like you can't count an infinite number of toys by taking them out one by one, you can't count an infinite number of real numbers by listing them out or taking them out one by one. Cantor's first uncountability proof showed us that some things are just too big to be counted in this way!