“Countably infinite” means an infinitely-large set of numbers that could be generated by infinitely following an algorithm with a finite number of steps. For example, natural (positive whole) numbers are countably infinite because they could be generated by following this simple algorithm:
Start with the number 1
Add 1 to your number
Repeat step 2
The set of real numbers, on the other hand, is uncountably infinite because you can have an infinite number of digits after the decimal place. You can’t define a finite generation algorithm like the one above simply because any precision you use wouldn’t cover the full range. In other words, if you wanted to modify the above algorithm, and chose 0.1 as your starting number, your algorithm would miss 0.01. If you chose to start at 0.01, you would miss 0.001, and so on
There are different kinds of infinity
“Countably infinite” means an infinitely-large set of numbers that could be generated by infinitely following an algorithm with a finite number of steps. For example, natural (positive whole) numbers are countably infinite because they could be generated by following this simple algorithm:
The set of real numbers, on the other hand, is uncountably infinite because you can have an infinite number of digits after the decimal place. You can’t define a finite generation algorithm like the one above simply because any precision you use wouldn’t cover the full range. In other words, if you wanted to modify the above algorithm, and chose 0.1 as your starting number, your algorithm would miss 0.01. If you chose to start at 0.01, you would miss 0.001, and so on