[Badiou and Science] 1.3 The Continuum Problem
“The impasse of ontology — the quantitative un-measure of the set of parts of a set — tormented Cantor… the multiple p(ωₒ) is the ontological schema of the geometric or spatial continuum. Demonstrating the continuum hypothesis, or (when doubt had him in its grips) refuting it, was Cantor’s terminal obsession: a case in which the individual is prey, at a point which he believes to be local or even technical, to a challenge of thought whose sense, still legible today, is exorbitant. For what wove and spun the dereliction of Cantor the inventor was nothing less than an errancy of being…” — Alain Badiou, Being and Event, pg. 295
We have rendered the idea of infinity a pretty banal concept in the Axiom of Infinity that creates ωₒ: being nothing fancy or mystical, the first infinity is just the set of all the ordinal numbers. A natural question to then ask is whether there are “more” infinities than just this set of ordinals?
Number Types
Here is a good place to review the major “types” of numbers many of us learned when we were young, and that we typically use as scientists interrogating the real world:
Whole numbers: e.g. 1, 27, 398, 4000, 59
Rational numbers — can be represented as a ratio of whole numbers: e.g. 8/1, 3/4, 2/3, 1/10000
Irrational numbers — cannot be represented as a ratio of whole numbers: e.g. π, √2, e
Real numbers: all rational and irrational numbers
Note that the whole numbers are a subset of the rational numbers, and that rational and irrational numbers make up the real numbers. Diagrammatically:
Denumerable Sets
As we previously discussed, denoting the size or “cardinality” of finite sets (such as [2 8 44 398]) and that of infinite sets (such as ωₒ) differs: the cardinality of finite sets is given by the ordinal numbers (e.g. the cardinality of [2 8 44 398] is 4, as it has four elements), while the cardinality of infinite sets in denoted by א (e.g. the cardinality of ωₒ is אₒ).
What will appear to be counter-intuitive at first glance is that the infinite cardinality אₒ is also the size of all “proper infinite subsets” of ωₒ. What this means is that the set of natural numbers (the ordinals) will have the same “size” as the set of, say, all the odd numbers or all the prime numbers:
In this way, the set of natural numbers and the set of all odd numbers (or all even numbers, or all prime numbers) can all be said to be “infinite sets” of the same size, with cardinality אₒ. Any set that can be put into “one-to-one correspondence” with the set of ordinal numbers has this property. Again, this is counter-intuitive (we expect there to be “more” counting numbers than even numbers) but this is a fundamental relation in set theory. We can “count” the elements of these infinite subsets of ωₒ (e.g. evens, odds, primes) using the ordinal numbers. Sets that can be counted using the ordinals in this way are called “denumerable” or “countable” sets.
What, then, can we say about the set of real numbers? Can they also be put into a one-to-one correspondence with the natural numbers? Is the set of real numbers denumerable?
The Continuum Hypothesis
Cantor’s “Continuum Hypothesis” originally arose from the simple question of “how many points are on a line?”, or equivalently, “how many real numbers exist?”. The linear “continuum” is thought to consist of all our empirical numbers, which are all the real numbers. It is intuitive to us that the size of this set will be “bigger” than the set of just the ordinals — after all, the real numbers consist of all the ordinals and the other rational numbers and the irrational numbers. Therefore, we may intuit that there are “more” real numbers than ordinal numbers, and that the real number set is not denumerable:
What is remarkable is that Cantor was not only able to prove that the cardinality of the set of reals is greater than that of the ordinals, but also that the cardinality of the reals is in fact equal to the cardinality of the power set of the ordinals; that is, Cantor was able to successfully demonstrate that the size of the set of real numbers (again, comprised of all the rational numbers and irrational numbers) that forms the linear continuum equals the size of the set of all subsets of the natural numbers (= the size of the power set of ωₒ).
This was done through Cantor’s “First Uncountability Proof” and his celebrated “Diagonal Argument’. I will leave interested readers to pursue the details of these proofs on their own, but the result of these breakthroughs was that Cantor believed he had solved part of the ancient continuum question: the cardinality of the real numbers — the very size of the linear continuum — precisely equals the cardinality 2^אₒ [1]:
What is now clear is that the set of real numbers is “bigger” than ωₒ, and therefore it must be of a size “larger” than our initial infinite size אₒ. But how much larger?
Just as in the finite realm ∅ is succeeded by [∅] (or 0 is succeeded by 1) with no ordinal remainder between the two, the infinite cardinal אₒ should be succeeded by the “next” infinite size of אₗ with no remainder in-between; the implication being that, if this were not the case, there would be “extra” numbers “between” the ordinals and the reals unaccounted for by the theory.
Therefore, the crux of Cantor’s “Continuum Hypothesis” rests on the following equation: 2^אₒ must equal אₗ, the next theoretical infinite measure. The proof of such an equation would represent the totality of all real numbers with no remainder between the amount of natural numbers (אₒ) and the amount of real numbers (אₗ), thus achieving the goal of a well-ordered, structured universe of numbers apprehended completely by thought alone:
The Continuum Problem
But can the above equation be proven? If not, it would suggest that the size of the continuum can be identified with any other infinite measure; that is, all we know is that (2^אₒ)> אₒ, but by how much? If (2^אₒ)≠ אₗ, then the continuum cannot be said to have an upper limit on its cardinality, and infinite disorder would reign — untold of possibilities of sets would be possible between the natural numbers and real numbers. 2^אₒ could possibly equal any infinite cardinal:
So, instead of binding the universe of numbers to the demands of order and structure, Cantor had unexpectedly unleashed the possibility of an infinite amount of infinities [2][3].
Cantor’s ultimate inability to prove or disprove his Continuum Hypothesis eventually drove him mad and led him to suicide. The Pandora’s Box he opened led to the infamous “Continuum Problem”: Is the size of the power set of the natural numbers — the very continuum — equal to the second infinite cardinal? Is (2^אₒ)= אₗ?
We will take leave the Continuum Problem for now, but will return to it later in greater detail, as historical attempts to solve the problems it brought up for the foundations of mathematics form the core of what Badiou describes as Situations and Events. For the moment, it suffices to recognize the potential for not just one or two types of infinite sets but many — the implication being the possibility of wild numbers “in excess” of the real numbers.
Is there a theory that describes such types of number not captured by our real numbers, and that would lend credence to this notion of multiple infinities? Stay tuned!
- Dr. G
Next: Surreal Numbers, a Trilogy
Notes
[1] This derivation primarily involves Cantor’s “diagonal argument”. See: Mary Tiles, The Philosophy of Set Theory, 107–111 and/or Howard DeLong, A Profile of Mathematical Logic, 76–81
[2] Kurt Gödel, “What is Cantor’s Continuum Problem?”, The American Mathematical Monthly (1947), 54, 515–25
[3] Mary Tiles, The Philosophy of Set Theory, Ch. 5
