38 | New Scientist | 16 April 2022
Features Cover story
The never-ending
infinity story
Mathematicians might have finally made a
breakthrough on a problem that has been baffling
them for 150 years, says Timothy Revell
I
NFINITY is a concept that is easy to
think about, but hard to understand.
Who hasn’t looked up at the night sky
and wondered whether space goes on forever?
Is it an endless expanse, or does it eventually
just stop? What does it mean if it doesn’t?
For trained mathematical brains, the
infinite is if anything even more bamboozling.
Mathematicians have known for well over a
century now that infinity isn’t just one thing,
it is infinitely many. There is an unending
tower of ever greater infinities stretching up all
the way to ... well, whatever you’d like to call it.
That isn’t even the worst of it. Although the
existence of this tower of infinities is a logical
consequence of mathematics as we know
it, that same mathematics is powerless to
describe it completely. Chip away at the
plaster to reveal the structure underneath and
you see that crucial load-bearing beams are
missing in the lower levels, suggesting that the
foundations of mathematics itself are unstable.
Mathematicians have long argued about
how best to shore the infinite tower up. Some
say we should simply leave well alone and
hope for the best. Others have proposed fixes,
variously deemed too costly, unlikely to
work or not in keeping with the original
style. No one has yet made anything like a
breakthrough. Except, perhaps, until now.
After decades of apparent stalemate, serious
progress seems to have been made on the
baffling question that lies at the heart of it all:
a nearly 150-year-old unproven conjecture
known as the continuum hypothesis.Humans have probably been thinking
about the unending for most of our existence.
“Infinity is a very natural concept,” says Vera
Fischer at the University of Vienna in Austria.
Jain mathematicians in India in the 4th and
3rd centuries BC believed that infinities
come in more than one size, but it wasn’t
until the 19th century that mathematician
Georg Cantor really started to grasp infinity’s
true, slippery nature.
To get a handle on his thought process,
imagine drawing a number line. The first
numbers you might add to it would be
the natural numbers – the counting
numbers that go 1, 2, 3 and so on. Although
mathematically imprecise, the “and so on”
means you could continue the counting
process forever. You will never run out of
natural numbers; their number is infinite.
That is where the weirdness starts, however.
Now think of the even numbers: 2, 4, 6 and so
on. Intuitively, you would say there are fewer
even numbers than there are natural numbers- half as many, perhaps. But that “and so on”
makes plain that there is no end point to them,
either. In fact, you can pair up every natural
number with an even number and vice versa –
(1,2), (2, 4), (3, 6) and so on – so there must be
the same “amount” of each. These two infinite
sets of numbers have the same size. This size is
written Յ 0 (pronounced aleph-null) and these
sets are said to be “countably” infinite.
There are lots of countable sets of numbers.
The integers, for example, which comprise
the natural numbers, zero and all the negative