I am trying to explain to someone the difference betwen countable and uncountable. I am not exactly 100% sure myself and a strictly mathematical definition doesnt help me. So now when I was thinking about it I came up with this explanation and Im wonder if it is accurate? a way to look at the difference betwen a countable and uncountable set would be to imagine you are number 4. If you are contained in a countable set you can look to your right and find for instance 4.1 and you can look to your left and find for instance 3.9. There are discrete values around you because 4.05 or 3.99 are not allowed. That means you can pinpoint the exact number that is closest to you and can count your way to any allowed number. You can jump to 4,1 and then 4,2 ect and count each step untill you reach any desired number. If you are contained in a uncountable set however you can look to your right and find no nearest number. If I say 4.1 is the nearest its not true because 4.01 is even closer and then we have 4.001 and so on to infinity, you have no closest neightboor, just a bunch of numbers that get infinitly closer without end. No discrete values are around you. So you can not jump count your way to 4.1 since there are just no defined places to jump to. Its impossible to count your way to any number and that makes it uncountable.