The concept of Paradox is critical to the development of mathematics as paradoxes highlight those areas of our knowledge where intuition or ‘common sense’ seems to fail.
The relationship of mathematics with reality has often been problematical. Mathematics has often been considered to be the purest of abstract sciences. The concepts and structures are largely defined without any need for a reality: they are logical entities and are self-contained (In its simplest form 1+1=2 without any need for there to be 2 of anything).The link between mathematics and reality is in the use of mathematics as a measuring tool, a counting tool or a predictive tool.
However there are times when the logic seems to fail and where intuition and the use of mathematics in reality is in doubt. From these illusions of failures we get our paradoxes, which require further mathematical investigation.
What is this thing called ‘Zero’?
Counting numbers (also called natural numbers) are the numbers 1, 2, 3, 4,.... etc. In linking numbers to reality the number zero (‘0’) is often included to complete the counting system. It is needed to explain the result of questions like:
'If we have 6 oranges and we eat them all, how many oranges are left? Without the number 0 we cannot answer this question’.
The natural numbers are also called positive integers, with their counterpart the negative integers: -1, -2, -3, -4, etc. Together they form the set of all integers - as long as we add the number zero between them. Zero is the only integer that is non-positive as well as non-negative. In this instance zero is a place marker: it is required to ensure the value between the integers remain correct. In the real world if you need to know the distance between 1 inch above a point on a wall (+1) to 1 inch below the point (-1) the answer is two (calculated as (+1)-(-1) =2). Two inches need to separate +1 inch from -1 inch, so place marker must exist - the number zero.
But the main paradox around the number zero is in division: it just doesn't work. Division is defined as the inverse of multiplication: for example if (8 = 4*2), then dividing by 2 on both sides gives (8/2 = 4). Similarly any number multiplied by 1 (the identity) gives the number itself: (5 = 1*5), and so with division we get (5/5 = 1).
With zero this doesn't work. If (0=1*0) then division by zero will leave us with (0/0 = 1). But (0 = 8*0) so (0/0 = 8). And so for all numbers it can be proven that it equals 0/0. Division is just not defined. The number system does not work if zero is treated as a number.
What is this thing called Infinity?
Infinity is actually not a number, but a concept. It doesn’t mean the ‘largest number’, for if you have a largest number then just add 1 to it, and you end up with a larger ‘largest’ number. Infinity roughly means the limit as you go higher and higher.
But funny things happen with infinity. Let us assume you can count all the positive numbers from 1 to infinity: 1, 2, 3, 4, …. Split all the positive numbers into 2 piles: a pile for the odd numbers and a pile for the even numbers. Set theory, a branch of mathematics, proves that the number of even numbers equals the number of odd numbers, each of which equals the total number of all the numbers we had to start with.
This strangeness of the behaviour of infinity is further demonstrated in Hilbert’s Paradox of the Grand Hotel. The size of the hotel is infinite, and the rooms are numbered, starting from 1 and rising: 1, 2, 3, 4, …. Now it is the busy period and every room is full. But a new guest arrives and begs for a room. What to do?
The hotelier assures the new guest he will find a room for him and he does so quite simply. He moves the guest from room 1 into room 2, the guest from room 2 into room 3, and so on. For whatever number room you can think of, he moves the guest from that room into the next one up. Infinity has no limit.
Mathematics has many paradoxes and each one teaches us a bit more about the logic behind the mathematical construct. The paradox often comes from applying this to reality. For example there is no ‘infinity’ in reality, only very large numbers (infinitely large, infinitely many). Some say there is a highest number – the number of particles in the galaxy, but mathematics, the ‘pure’ science refutes this.
I have not proven anything rigorously, nor have I correctly defined Infinite in the discussion of “Countably Infinite”. If I have made any specific errors other than simplification, I apologise. No specific sources were used, though there are discussions of the concepts in Wikipedia.