Computably enumerable set

Okay, so let's imagine we have a really big pile of toys. But instead of toys, these are actually numbers. And we want to figure out which numbers are in this big pile.

A computably enumerable set is like someone giving us a checklist and saying "hey, instead of digging through this whole pile, just look at this list and find the numbers on it."

But how did they make this list? They did it by using a computer program that follows a very specific set of rules. These rules make sure that every number in the pile will eventually get put on the list, but sometimes it takes a really long time.

It's kind of like if we had a really big book and we wanted to make a list of all the words in it. We could use a computer program that would read through the book and write down every word it finds. But since the book is so big, it might take the program a really long time to finish making the list.

So a computably enumerable set is like a list of numbers that was made using a computer program that follows certain rules to make sure it includes all the numbers in a really big pile.