Logics for computability

Imagine you have a special box that takes in different numbers, and it tells you if each number is "yes" or "no." This box is like a computer program, and it uses logic to figure out if the numbers are "yes" or "no."

Now, there are some numbers that this box can't figure out if they are "yes" or "no." These are called "undecidable" numbers. There is no logic that can always figure out if these numbers are "yes" or "no."

But there are some numbers that the box can always figure out if they are "yes" or "no." These are called "decidable" numbers. There is a logic that can always figure out if these numbers are "yes" or "no."

The study of logics for computability is all about figuring out which numbers are decidable and which ones are undecidable. It's like playing a game of "yes or no" with the box, and trying to figure out the rules that it follows to decide if a number is "yes" or "no." It's a really important part of computer science and math, because it helps us understand what kinds of problems computers can solve, and what kinds they can't.