The Different Sizes of Infinity
“The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds” – Georg Cantor
Infinity is to us what light is to a moth, we’re drawn to it and yet the concept of it holds so much mystery and power that sometimes we forbid ourselves from involving ourselves with it. It’s as if we religiously abstain from it, infinity must be respected above all else and we should not try and probe it.
But the mathematics of infinity is an extremely developed field. Mathematicians have constructed all kinds of different infinities of various sizes and characters. The real mystery is whether infinity in all its various guises is actually something that is realised in the physical world.
Is the universe infinite in extent? Is spacetime infinitely continuous on the smallest of scales, or is spacetime made of discrete chunks like bits in computer code? Are black holes infinitely dense, or are they do they just have huge, but finite, densities?
Infinity seems so incomprehensible to us, we’re so unable to experience it, that we believe it cannot ever be realised in nature. The infinities that plagued Quantum Field Theories were seen as flaws of the theory – no calculation should yield an infinite answer.
But who are we to say that nowhere in nature does infinity appear? Why should we exclude its appearance in our physical theories. There’s a wealth of mathematical research that has been conducted concerning infinity, yet virtually none of it is actually used in physical theories.
String theory, M-Theory and all of the candidate “Theories of Everything” are formulated in terms of highly sophisticated mathematics – group theory, algebraic topology, Clifford algebra, to name but a few, but no theory of physics utilises any of the ideas studied in the mathematics of the infinite. Perhaps this is a field that theoretical physicists should start looking into?
In any case, here’s a taste of some of the things we know about infinity:
Infinity has Different Sizes
Consider the integers, or “natural numbers” 1,2,3,4,5,6,7,…. how many of them are there? We say that there is a “countably infinite” number of integers, and we use a symbol called ‘aleph null’ to represent it:
Aleph null is the smallest size of infinity that we know about, it’s probably safe to associate this kind of infinity with the traditional symbol – the sideways 8.
We say that any set of objects that we can label using the integers has the same size as the integers. Take the even integers for example. Although it may seem that the number of even integers should be exactly half the number of integers, the set of even numbers actually has the same size. This is because we can label each of the even numbers with an element from the set of integers:
2 is labelled with 1
4 is labelled with 2
6 is labelled with 3
,,, and so on
Because we can carry out this labeling (called a one-to-one correspondence, or bijection) we say that there are aleph null even numbers, the same number as integers!
If we consider the real numbers however – all the decimals – it turns out that there are more of them than integers. The most elegant proof of this was written down by Georg Cantor, he showed that however you tried to label the decimals with integers, there would always be a decimal that you will have missed.
The number of real numbers is thus larger than the number of integers – there’s an infinity larger than the infinity of integers!
The number of real numbers is denoted
But there’s more…
As the name “aleph nought” suggests, we can construct larger infinite numbers. Aleph nought is the smallest, followed by aleph one, aleph two, and so on. These are called Aleph numbers and they represent the number of some really strange mathematical beasties called ordinal numbers.
Aleph one is the number of all countable ordinal numbers, it is itself an ordinal number larger than all countable ones, so it is an uncountable set making it distinct from aleph null and not just a made up concept.
But mathematicians have taken this all to dizzying heights, there’s a whole zoo of different infinities that have been concocted on top of these ones:
- Beth Numbers
- Inaccessible Cardinals
- Strongly Unfoldable
- Ramsey Numbers
- Woodin Numbers
- Almost Huge
- Ethereal Cardinals
- Ineffable Cardinals
Perhaps my favourite thing about these infinities is just how many of them there are and how cool their names are!
Infinity is still an extremely mysterious concept that we’re unable to properly grasp, yet mathematicians have penetrated deeper into the world of infinity than ever before, and there is a wealth of new discoveries. The interesting question is whether these infinities have any relevance to the real physical world.