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.*

*This article was originally published in our “Uncertainty” issue in June, 2013.*