The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy
Consider the following sentence:
“This statement is false.”
Is that true?
If so, that would make
this statement false.
But if it’s false, then the statement
is true.
By referring to itself directly, this
statement creates an unresolvable paradox.
So if it’s not true and it’s not false—
what is it?
This question might seem
like a silly thought experiment.
But in the early 20th century,
it led Austrian logician Kurt Gödel
to a discovery that would change
mathematics forever.
Gödel’s discovery had to do with
the limitations of mathematical proofs.
A proof is a logical argument
that demonstrates
why a statement about numbers is true.
The building blocks of these arguments
are called axioms—
undeniable statements
about the numbers involved.
Every system built on mathematics,
from the most complex proof
to basic arithmetic,
is constructed from axioms.
And if a statement about numbers is true,
mathematicians should be able to confirm
it with an axiomatic proof.
Since ancient Greece,
mathematicians used this system
to prove or disprove mathematical claims
with total certainty.
But when Gödel entered the field,
some newly uncovered logical paradoxes
were threatening that certainty.
Prominent mathematicians were eager
to prove
that mathematics had no contradictions.
Gödel himself wasn’t so sure.
And he was even less confident
that mathematics was the right tool
to investigate this problem.
While it’s relatively easy to create
a self-referential paradox with words,
numbers don't typically
talk about themselves.
A mathematical statement is simply
true or false.
But Gödel had an idea.
First, he translated mathematical
statements and equations into code numbers
so that a complex mathematical idea could
be expressed in a single number.
This meant that mathematical statements
written with those numbers
were also expressing something about
the encoded statements of mathematics.
In this way, the coding allowed
mathematics to talk about itself.
Through this method, he was able to write:
“This statement cannot be proved”
as an equation,
creating the first self-referential
mathematical statement.
However, unlike the ambiguous
sentence that inspired him,
mathematical statements must be
true or false.
So which is it?
If it’s false, that means the statement
does have a proof.
But if a mathematical statement has
a proof, then it must be true.
This contradiction means that Gödel’s
statement can’t be false,
and therefore it must be true that
“this statement cannot be proved.”
Yet this result is even more surprising,
because it means we now have
a true equation of mathematics
that asserts it cannot be proved.
This revelation is at the heart
of Gödel’s Incompleteness Theorem,
which introduces an entirely new class
of mathematical statement.
In Gödel’s paradigm, statements still
are either true or false,
but true statements can either be
provable or unprovable
within a given set of axioms.
Furthermore, Gödel argues these
unprovable true statements
exist in every axiomatic system.
This makes it impossible to create
a perfectly complete system
using mathematics,
because there will always be true
statements we cannot prove.
Even if you account for these
unprovable statements
by adding them as new axioms
to an enlarged mathematical system,
that very process introduces new
unprovably true statements.
No matter how many axioms you add,
there will always be unprovably true
statements in your system.
It’s Gödels all the way down!
This revelation rocked the foundations
of the field,
crushing those who dreamed that every
mathematical claim would one day
be proven or disproven.
While most mathematicians accepted this
new reality, some fervently debated it.
Others still tried to ignore
the newly uncovered a hole
in the heart of their field.
But as more classical problems were proven
to be unprovably true,
some began to worry their life's work
would be impossible to complete.
Still, Gödel’s theorem opened
as many doors as a closed.
Knowledge of unprovably true statements
inspired key innovations
in early computers.
And today, some mathematicians dedicate
their careers
to identifying provably
unprovable statements.
So while mathematicians may have
lost some certainty,
thanks to Gödel they can embrace
the unknown
at the heart of any quest for truth.
