Georg Cantor died in
1918 in a sanatorium in Halle, Germany. A pre-eminent mathematician, he had
laid the foundation for the theory of infinite numbers in the 1870s. At the
time, his ideas received hostile opposition from prominent mathematicians in
Europe, chief among them Leopold Kronecker, once Cantor’s teacher. In his first
known bout of depression, Cantor wrote 52 letters to the Swedish mathematician
Gösta Mittag-Leffler, each of which mentioned Kronecker.
But it was not just
rejection by Kronecker that pushed Cantor to depression; it was his inability
to prove a particular mathematical conjecture he formulated in 1878, and was
convinced was true, called the Continuum Hypothesis. But if he blamed himself,
he did so needlessly. The debate over the conjecture is profoundly uncertain:
in 1940 Kurt Gödel proved that the Continuum Hypothesis cannot be disproven (technically speaking, that the negation of the Hypothesis cannot be proven),
and in 1963 Paul Cohen proved that it cannot be proven. Poor Cantor had chosen
quite the mast to lash himself to.
In his first known bout of depression, Cantor wrote 52 letters to the Swedish mathematician Gösta Mittag-Leffler, each of which mentioned Kronecker.
How is it possible,
though, for something to be provably neither provable nor disprovable? An exact
answer would take many pages of definitions, lemmas, and proofs. But we can get
a feeling for what this peculiar truth condition involves rather more quickly.
Cantor’s Continuum
Hypothesis is a statement regarding sizes of infinity. To see how infinity can
have more than one size, let’s first ask ourselves how the sizes of ordinary
numbers are compared. Consider a collection of goats in a small forest. If
there are six goats and six trees, and each goat is tethered to a different
tree, then each goat and tree are uniquely paired. This pairing is called a
“correspondence” between the goats and the trees. If, however, there are six
goats and eight trees, we will not be able to set up such a correspondence: no
matter how hard we try, there will be two trees that are goat-free.
Correspondences can
be used to compare the sizes of much larger collections than six goats—including
infinite collections. The rule is that, if a correspondence exists between two
collections, then they have the same size. If not, then one must be bigger. For
example, the collection of all natural numbers {1,2,3,4,…} contains the
collection of all multiples of five {5,10,15,20,…}. At first glance, this seems
to indicate that the collection of natural numbers is larger than the
collection of multiples of five. But in fact they are equal in size: every
natural number can be paired uniquely with a multiple of five such that no
number in either collection remains unpaired. One such correspondence would
involve the number 1 pairing with 5, 2 with 10, and so on.
If we repeat this
exercise to compare “real” numbers (these include whole numbers, fractions,
decimals, and irrational numbers) with natural numbers, we find that the
collection of real numbers is larger. In other words, it can be proven that a
correspondence cannot exist between the two collections.
The Continuum
Hypothesis states that there is no infinite collection of real numbers larger
than the collection of natural numbers, but smaller than the collection of all real
numbers. Cantor was convinced, but could never quite prove it.
To see why, let’s
begin by considering what a math proof consists of. Mathematical results are
proven using axioms and logic. Axioms are statements about primitive
mathematical concepts that are so intuitively evident that one does not
question their validity. An example of an axiom is that, given any natural number (which is a primitive concept), there exists a larger natural number. This is
self-evident, and not in serious doubt. Logic is then used to derive
sophisticated results from axioms. Eventually, we are able to construct models,
which are mathematical structures that satisfy a collection of axioms.
Crucially, any
statement proven from axioms, through the use of logic, will be true when
interpreted in any model that makes those axioms true.
If there are six goats and eight trees, we will not be able to set up such a correspondence: no matter how hard we try, there will be two trees that are goat-free.
It is a remarkable
fact that all of mathematics can be derived using axioms related to the
primitive concept of a collection (usually called a “set” in mathematics). The
branch of mathematics that does this work is known as set theory. One can prove
mathematical statements by first appropriately interpreting the statement in
the language of sets (which can always be done), and then applying logic to the
axioms of sets. Some set axioms include that we can gather together particular
elements of one set to make a new set; and that there exists an infinite set.
Kurt Gödel described
a model that satisfies the axioms of set theory, which does not allow for an
infinite set to exist whose size is between the natural numbers and the real
numbers. This prevented the Continuum Hypothesis from being disproven.
Remarkably, some years later, Paul Cohen succeeded in finding another model of
set theory that also satisfies set theory axioms, that does allow for such a set to exist. This prevented the Continuum
Hypothesis from being proven.
Put another way: for
there to be a proof of the Continuum Hypothesis, it would have to be true in
all models of set theory, which it isn’t. Similarly, for the Hypothesis to be
disproven, it would have to remain invalid in all models of set theory, which
it also isn’t.
It remains possible
that new, as yet unknown, axioms will show the Hypothesis to be true or false.
For example, an axiom offering a new way to form sets from existing ones might
give us the ability to create hitherto unknown sets that disprove the
Hypothesis. There are many such axioms, generally known as “large cardinal
axioms.” These axioms form an active branch of research in modern set theory,
but no hard conclusions have been reached.
The uncertainty
surrounding the Continuum Hypothesis is unique and important because it is
nested deep within the structure of mathematics itself. This raises profound
issues concerning the philosophy of science and the axiomatic method.
Mathematics has been shown to be “unreasonably effective” in describing the
universe. So it is natural to wonder whether the uncertainties inherent to
mathematics translate into inherent uncertainties about the way the universe
functions. Is there a fundamental capriciousness to the basic laws of the
universe? Is it possible that there are different universes where mathematical
facts are rendered differently? Until the Continuum Hypothesis is resolved, one might be tempted to conclude
that there are.
Ayalur Krishnan is an
Assistant Professor of Mathematics at Kingsborough CC, CUNY.
