Intuitively, a finite universe seems much more reasonable to us than an infinite one, if for no other reason than because in an infinite universe (or multiverse) everything that could happen, would happen.
What is the largest number?
A google is a 1 with a 100 zeros after it. This is higher than the atoms in the observable universe.
A googleplex is 1 to the power of a google. You could not write this number down in the observable universe even if the you used a separate atom for each zero.
But far far larger than this is Graham’s number, named after Ron Graham. This is larger in proportion to a googleplex than a googleplex is to the number 10.
Still, as a finite number this is no closer to infinity than 1.
What is infinity?
It is a number the digits of which never come to an end.
The first paradox about infinity is that an infinite list of even integers (whole numbers) is the same size as an infinite list of all integers. This is because every number of the even list can be paired with the next number in the general list.
Almost 100 years ago the German mathematician David Hilbert used the example of a hotel with infinite rooms to illustrate how baffling the second paradox is: that infinity + 1 is still infinity. As such, there is always an extra free room in a completely full hotel with infinite rooms. A variation of this is that infinity + infinity (or infinity x 2) is also infinity because each guest already there would move into the room with the number double that of their original one. And with subtraction, because infinity – 3 is still infinity, infinity minus infinity could be greater than zero. In the case of that example it would be 3.
Aristotle thought that an infinite number made sense but that it was impossible to incorporate this into a logical system. Of course he was not a mathematician, but this did lead to his belief that time (though not space) was infinite in both directions.
Contemporary mathematician Doron Zielberger says that infinity is repulsive and a fiction. His view is that when you add 1 to the biggest number you get back to zero. Of course we could never count to this biggest number, so his reason for believing this isn’t direct evidence, but because it is more reasonable than taking on board the paradoxes of infinity. But isn’t imagining a biggest number just as difficult?
In the late 19th and early 20th century Georg Cantor was the first to develop a proper mathematics of infinity, using a branch of mathematics called set theory. His great discovery was that the infinity of real numbers (decimal numbers) is larger than the infinity of integers. He proved this using his diagonal argument: given any list of real numbers we can create a new one not on that list by counting down between them diagonally (i.e. taking the first digit from the first number, the second from the second, the twenty first from the twenty first etc). This shows that the list of real numbers cannot be paired on a one to one basis with the list of integers, which proves that the former is larger. Mathematicians call the set of all integers ‘countably infinite’ and the set of all real numbers ‘uncountably infinite’.
Towards the end of his life Cantor went insane, though this was more influenced by the then under-appreciation of his work, than by the trials of his work itself. His work is fundamental to contemporary mathematics because infinity is quicker and easier to manipulate than large finite numbers.
Is the universe infinite?
It looks like space is infinite because we don’t see any edge or boundary to it.
If the universe is infinite then a monkey would be able to accidentally type the complete works of Shakespeare. This is as unlikely as the same person winning the (British) National Lottery every week for 29,000 years. Yet it would not only happen, but it would happen an infinite number of times.
More worryingly, there will be planets exactly like Earth on which live people exactly like ourselves. These counterparts of ourselves would have the exact same names and memories as us! The distance between our world and theirs would be untraversable by a human, but they would still be there … an infinite number of them!!!!
Yet of course there is no need to believe that the universe is infinite because it has no edge, because space time could well be folded back on itself as many contemporary scientists believe.
There is a need to explain how such distant parts of the universe could have such uniform cosmic microwave background temperature. This need lead to the development of inflationary cosmology. This predicts that the universe could be infinite because it will carry on expanding.
At the end of the documentary one scientist suggested that the universe is not only infinite but that the big bang coincided with the creation of an infinite number of other universes separate from our own, all of which are themselves infinite. He made the astonishing claim that this was our best model of cosmology today, but didn’t say anything towards how or why this was the case, nor indeed show any endorsement for his extravagant theory.