16 April 2022 | New Scientist | 41GR
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Mathematics needs infinity. Two of
the most practically useful branches
of mathematics, trigonometry and
calculus, use it, but in such a way that
the infinities are conveniently hidden
from the real world. You need the
infinities involved in both to work out
the optimum path of a rocket, but
neither the trajectory nor the velocity,
or any practically measurable quantity,
will ever come close to the infinite.
Mathematically, infinity is useful, but
does it really exist in the physical world?
In many theories of physics,
the appearance of actual infinities is a
sign of things going awry. In Einstein’s
equations of general relativity, for
instance, the “singularity” at the centre
of a black holes is an infinitely warped
point in space-time; similarly, wind
the clock of our expanding, cooling
universe back some 13.8 billion
years and you reach the point of the
big bang where it is supposedly
infinitely dense and hot. Actually,
however, we know that general
relativity breaks down when we
reach these tiny, extreme scales
where quantum theory also comes
into play. The infinities seem to be the
equivalent of a mathematical shrug:
don’t know what’s going here, sorry.
The quantum field theories that
underlie the standard model of particle
physics provide another example.
These were once filled with
nonsensical predictions that, for
example, the mass and charge of an
electron were infinite. Over decades,
physicists have steadily managed
to remove many of these infinities,
creating today’s highly successful
model. If the standard model can one
day be unified with general relativity
to give a complete picture of the
universe, perhaps the infinities will
disappear completely – or of course,
perhaps they won’t.IS INFINITY
REAL?
Georg Cantor
discovered that
infinity comes
in infinite
varietiesmathematicians care about,” says David
Asperó at the University of East Anglia, UK.
It covers everything from building the real
numbers to constructing a working theory
of arithmetic. The hunch was that these
foundations just weren’t strong enough to
bear the full weight of the infinite tower.
One of the first proposals for a new
supporting axiom emerged from a technique
called forcing, introduced by Cohen in the
1960s. Loosely, Cohen started with a set of the
real numbers of size Յ 1 – with no assumption
that this equated to the continuum infinity
or not – and then used this technique to cram
more into it. Forcing was an extremely
powerful tool for building interesting new
mathematical sets. But it turned out that you
could end up with the real numbers being any
size of infinity – not just Յ 1 or the next one up,
Յ 2 , but Յ 42 or anything else. It was simply too
powerful a technique to say anything useful
about where the continuum infinity lay.
An answer seemed possible by restricting
what forcing could do. In 1988, mathematicians
Menachem Magidor, Matthew Foreman and
Saharon Shelah proposed an axiom called
Martin’s maximum, named after set theorist
Donald Martin, that could do exactly that. And
it gave, finally, an answer to the continuumhypothesis: that it was false. The continuum
infinity isn’t Յ 1 , but Յ 2 ; there is a level of infinity
between the countable and the continuum.
A decade later, in 1999, Hugh Woodin, then
at the University of California, Berkeley,
suggested another approach, called (*) and
pronounced “star”. Like forcing, this also
allowed mathematical objects to be built, but
in a slightly different way. It, too, said that the
continuum hypothesis is false, and that the
continuum infinity lies at the third level up, Յ 2.A new level of infinity?
There is no easy, intuitive example of what
might sit between the countable and the
continuum, but there are some complicated
constructions that we know would do whether
or not Յ 1 and the continuum are the same
thing. The Hausdorff gap, for example, is a set
involving sequences of numbers that has size
Յ 1 regardless of the size of the continuum.
Still, with two competing ways to fortify the
foundations of mathematics both giving the
same answer to the continuum hypothesis,
you might think it was game over. But then
came another twist that could be loosely
characterised as Woodin changing his mind.
Both (*) and Martin’s maximum are axioms >