Another apparently ridiculous assertion! After all, did I not state in my last post that infinity is a concept that describes something unbounded? So how can anything be greater than something that is unbounded? The answer is that it depends on the way it is unbounded. Lets consider the infinite list of all the positive numbers in sequence 1,2,3 etc. You know that the list is complete because you have not missed any integer. After all, by definition you have listed them in order (2 definitely follows 1 and 3 definitely follows 2 and so on). 

Now think of the infinite list of even numbers listed in sequence 2,4,6 etc. Again the infinite list of these numbers is complete since by definition you have listed them in order. You can now map each number on the first list to each number on the second (1 maps to 2, 2 maps to 4, 3 maps to 6 etc). There is no number on the first list that you cannot map to a number on the second list since any integer can be multiplied by 2 and every number that is divisible by 2 is on the second list. These lists are both unbounded but they are unbounded in the same way (because of this one to one mapping) and are known as “countable” lists. 

Now try and do the same thing with the positive real numbers. You write the first positive real number down ….umm, what are you going to write? Perhaps you think of 0.1 but then you know that 0.01 is less than that. So where are you going to start?  You literally cannot write down a list of all the real numbers in sequence as whatever number you start with you have missed out a number smaller than it! As a result you cannot possibly map each integer on our list of integers to a corresponding number on the list of real numbers because whatever real number you choose to map to the integer 1, there is another (actually an infinity of other) smaller real numbers that you could choose to map it to. So you see that the list of all real numbers is not even countable and therefore it is a bigger infinity than the list of all positive integers (or the infinite list of any other countable things). You think that must be it, but what is even more astonishing is that mathematicians have identified lists that are even more unbounded than the list of real numbers. Good grief, my head aches…time to stop!