The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite
~ Georg Cantor

Recently NETFLIX released a documentary on mathematical concept of infinity, titled A Trip to Infinity. NETFLIX’s trip is a bad one.

The first twenty minutes or so are reasonable but after that it descends into mysticism, sensationalism, speculation and mathematical nonsense. The average intelligent viewer will be left confused and intimidated and will have a poorer understanding of infinity than they had before watching Trip.

The worst mistake they make is referring to infinity in the singular, as if there is only one infinity. In fact in the first twenty minutes they make it clear that there is more than one, then return to talking about “infinity” instead of “infinities”.

Many infinities

When people talk about “infinity” they think first of the natural numbers and the set {0,1,2,3,…}. But there is also the set of points on the real line between 0 and 1. It’s not obvious but these infinities are not the same.

The study of infinities was initiated by Georg Cantor more than 100 years ago. He was the first to realize that there are many infinities and that they are linearly ordered by size – given two infinities one is at least as big as another.

He had a very simple rule for comparing infinite sets: S1 ≤ S2 (S2 has at least as many elements as S1) if there is a one-to-one comparison between S1 and a subset of S2. In particular, he proved that is S1 ≤ S2 and S2 ≤ S1, there is a one-to-one correspondence between elements of S1 and S2.

For example, if E is the set of even numbers and S the set S of square numbers then E ≤ S because the correspondence 2n ⟷ n² is a one-one correspondence between E and S. The reverse correspondence shows that S≤ E and so S≡E; the ‘number’ of squares is the same as the number of even numbers.

Similar arguments show that S and E have the same number if elements as {0,1,2,3,…} (this set is called 𝜔). In fact any infinite set whose elements can be enumerated has the same number of elements as 𝜔. The number of elements of a set is called its cardinality. The cardinality of 𝜔 is the smallest infinity and is called ℵ₀

Is it the only one? It’s not hard to see that any other infinity must be larger than ℵ₀.

This is where Trip goes off the rails. We need to compare 𝜔 and the set ℛ of real numbers.

The cardinality of ℛ

Cantor proved that the cardinality of ℛ is greater than ℵ₀ with a famous proof that has been modified and adapted many times.

Suppose we could enumerate the elements of ℛ. We could lay out the enumeration in a two dimensional table, like this

1 .3 1 4 1 5 ..
2 .2 7 1 8 2 …
3 .3 3 3 3 3 …
4 .1 4 1 5 9 …
5 .6 9 3 1 4 …
…

Now consider the diagonal number . 37354 … . This number might be somewhere in the list. But let’s take it and change each digit, say by adding 1 mod 10, giving .48465 … . This number can’t be in the list because it differs from the n_^th number in the n^th digit (we have to take care of repeating 9s). So the set of real numbers between 0 and 1 can’t have cardinality ℵ₀.

This is Cantor’s diagonal argument and it shows ℵ₀ < the cardinality of ℛ. There are at least two infinities. The cardinality of ℛ is called ℶ₁.

By this point Trip goes back to talking about “infinity” and the experts are staring at what looks like a billiard ball which its supposedly the universe …. or something.

Meanwhile back in mathland it gets even more interesting. ℵ₀ and ℶ₁ are good for very many sets. The set of pairs of natural numbers and in fact the set of finite sequences of natural numbers has cardinality ℵ₀. Thus there are only ℵ₀ many polynomials with integer coefficients.

The same is true of the set Q of rational numbers and the set of finite sequences of rational numbers. That means only ℵ₀ many polynomials with rational coefficients

Similar results hold for ℛ. There are ℶ₁ polynomials with real coefficients.

The cardinals ℶ₂, ℶ₃, ℶ₄, ,…

But now consider drawings on the plane. If every pair of rationals is a pixel which can be black or white, there are ℶ₁ possible images. And if every pair of reals is a pixel, there are more than ℶ₁ possible images.

How many more? Cantor’s diagonal argument can be generalized to show that the set of subsets of a set has a bigger cardinality than that of the set. If the set has cardinality 𝜅 then the powerset ℘(S) (the set of all subsets of S) has cardinality 2^𝞳, which is bigger than 𝜅.

An image on the plane is a subset of ℛxℛ and the set of such images has cardinality 2^ℶ₁. So now there are three infinities. (2^ℶ₁ is called ℶ₂)

We can continue taking power sets and generate a sequence ℶ₁, 2^ℶ₁ (=ℶ₂), 2^ℶ₂(=ℶ₃), … of bigger cardinalities. So there are at least ℵ₀ many infinities.

Nothing of this, which is mind boggling, is described in Trip. Instead they’re speculating about an orange in a box … which supposedly disintegrates then reassembles itself.

The continuum hypothesis

Back to the math, which really gets interesting. When Cantor discovered the power-of-two series ℶ he naturally started wondering if they were all the cardinalities. In particular, he wondered if there is a set of reals whose cardinality is greater than ℵ₀ but less than ℶ₁. (Incidentally ℶ₁ = 2^ℵ₀).

He tried for many years to settle the question but never succeeded, neither finding such a set nor proving that none exists. He was forced to leave what he called the “continuum hypothesis” (CH) unresolved.

It wasn’t till 1938 that any progress was made. The famous logician Kurt Gödel proved that it’s not possible to refute the continuum hypothesis. So it’s either true or else …

… the or else was demonstrated in 1963 by Paul Cohen. He showed that it is not possible to prove the continuum hypothesis either. In other words, CH is independent of the usual axioms of set theory (which are taken to be the axioms of modern math).

A lot of progress was made after Cohen’s proof but nothing decisive. It turns out that the axioms of math have little to say about the cardinalities between ℵ₀ (=ℶ₀) and ℶ₁. There could be a couple or ℵ₀ many or more.

This is interesting, even mind boggling, but none of it shows up in Trip. At this point they’re speculating that eventually the heat death of the universe will kill off humanity.

Constructibility

Since Cantor, mathematicians have been searching for a plausible extra axiom that will settle the continuum hypothesis. There is no consensus.

In my opinion, however, there is one obvious candidate, namely the axiom of constructibility.

This axiom, sometimes written “V=L”, says that every set is “constructible”. Roughly speaking, the constructible sets are those that are definable in terms of simpler constructible sets, and aren’t just introduced arbitrarily at random.

Gödel introduced the constructible sets to prove his partial independence results. V=L implies the general continuum hypothesis (ℵ_𝛂 = ℶ_𝛂 for all 𝛂) and the axiom of choice plus a whole lot of other results.

My championing of V=L is probably due to my computer science background, where we encounter many recursive definitions. As a general rule we take the meaning of a recursive definition X = f(X) to be the least fixed point of the equation. And we calculate the least fixed point by starting with nothing (∅) and iterating ∅,f(∅),f(f(∅)),f(f(f(∅))),… then taking the limit.

The ordinary axioms of math imply that any family F of sets is closed under definitions; so the family V of all sets satisfies the recursive definition V = ∅ ∪ 𝒟(V), where 𝒟(X) = sets definable from X. If we apply the iterative/cumulative procedure described above, we get L, the family of constructible sets.

I’m sure that this is the ‘right’ thing to do and that the axiom of constructibility should be considered as ‘true’.

Beware of Pop Science

So Trip to Infinity is bad news. Unfortunately it’s not alone. Trip veered into physics and almost every pop explanation of physics is just as bad.

Speculation, sensationalism, paradoxes, misinformation. Blackholes, wormholes, time travel and the like.

One frequently repeated example of false facts is the claim that bodies are collections of atoms with nothing in between. In reality the space between atoms is filled with fields: electromagnetic, gravitational, who knows what else. Every physicist knows this, but only too many are willing to go before the public and declare otherwise.

This rotten pop science is basically a plot to make people feel stupid. They’re intimidated because what they see doesn’t make sense and they conclude they’re not smart enough for science.

A good example in Trip comes when they discuss cardinal arithmetic (cardinals can be added, multiplied etc like integers). They show the equation ∞ + 1 = ∞, which is true if ∞ is an infinite cardinal,
then proceed to subtract ∞ from both sides, giving 0=1, a howling contradiction. And that’s where they leave it.

What is the viewer supposed to make of this? That infinity is a contradiction?

In fact, all it means is that not all the rules of finite arithmetic apply to cardinal arithmetic. No big deal. There’s nothing wrong with your brain and 0 is not equal to 1.

Pop science has tried to scramble your brains.