Countable vs uncountable sets

Countable sets are collections where you can list all items one by one, like whole numbers (1, 2, 3, …), even if the list is infinite. Uncountable sets are too large to list each element individually, like real numbers between 0 and 1, which form a continuous range. In essence, countable sets have a size that matches the natural numbers, while uncountable sets are bigger, meaning their elements can’t be matched one-to-one with the natural numbers. This distinction helps mathematicians understand different sizes and infinities within sets.