Refining the Axioms Set of Theory

Fair warning. This is an intense dive into the nuance of mathematical infinity. It’s mean to be read as a primer for Natalie Wolchover’s recent piece on the foundations of mathematics in Quanta Magazine. Like her piece, it may take more a simple scan to digest.

Infinity is nuanced.

Consider π, the ratio of a circle’s circumference to its diameter?

π = 3.1415926...

Exactly how many digits to pi are there? An infinite amount. You can count forever and distribution of digits in π will never repeat.

You might ask, how many numbers like π, the so-called irrational numbers, are there?

Infinitely many. Not only that, there are infinitely more irrational numbers than there are digits in π.

Sizes of Infinity
The Familiar infinites come in two sizes: discrete and continuous. We can define larger, more complicated versions of infinity, but when it comes to the real numbers, we usually only need these two. Let’s call them D(discrete) and C(continuous).

How many counting numbers are there? D
How many real numbers exist between 0 and 1? C
How many even numbers are there? D
Odd numbers? D
How many numbers between 0 and 3? C
Between 0 and 0.0001? C
How many fractions of finite whole numbers are there? D

That can be confusing, so let’s be clear. There are just as many odd numbers as there are natural numbers. Infinity is weird that way. There are just as many even, whole numbers are there are fractions of all the whole numbers. There are just as many whole numbers divisible by 29 as there are decimals in pi.

How could this be? Because you can line them all up, one to one, and count out to infinity. Yup, even with fractions (see the image at the top of the piece). That’s the problem and the beautiful simplicity with D-type infinites, they never, ever end.

Any continuous subset of the real numbers cannot be lined up with the whole numbers. There’s just too many of them. Put differently, the fractions are just tiny, isolated islands in a sea of irrational numbers with infinite, nonrepeating decimal expansions.

In other words, ANY infinite subset of the real numbers either has the size C - and looks like a continuous section of a line - or D - and looks like a discrete collection of points.

Despite this weirdness, infinity simplifies things: Two sizes fit all.

Or so we thought.

Axioms of Set Theory and their Implications
This distinction between infinities, D and C, is hopefully clear. But the actual size of C is something of a definition. Saying that there are only two sizes of infinite subsets of the real numbers is a choice. It’s an axiom of modern set theory.

Axiomatic set theory provides the foundation for modern mathematics. The current, commonly preferred set of axioms for set theory is independent of this choice. You could have three, four or ten layers of infinite in between. Proving that fact was highly nontrivial.

There’s only one problem. We have no idea what an intermediate, infinite subset would look like. The whole numbers give us an explicit model for D. And we already know how to build sets of size C from those of size D.

Here’s a tongue twister: the set of all possible subsets of a set of size D has the size C.

In other words, you can create a model for the continuum of real numbers using the set of all possible singles, pairs, triplets, quadruplets, etc. of whole numbers. This will also include all the infinite subsets like the even numbers, the odd numbers and those divisible by 137. This is called the Power Set of those natural, discrete, whole numbers.

So the question put forth by the Cantor’s famous Continuum Hypothesis is whether or not there is a subset of the Power Set whose size is larger than D, but somehow smaller than C. Removing one or two numbers doesn’t really help, infinity minus one or two still equals infinity. If this is confounding to you, you’re in good company. As things currently stand, there is no solution. The answer requires an axiom!

But what if it weren’t so?

New Approaches to Axiomatic Set Theory
In her recent piece in Quanta, Natalie Wolchover writes about two independent attempts to study infinity by augmenting the currently established “Zermelo-Frankel-Choice” axioms of set theory.

Why would you do this? Well, it’s interesting to see what doesn’t change when you change the subtle rules of the game. The natural numbers are still there. The real numbers are still there. You can still model the continuum with the power set of the natural numbers. In some sense, these ideas are stable against changing axioms: they’re far more important than the mere axioms themselves.

What’s curious is that different refinements of the rules have led to the same outcome. Both refinements discussed in Natalie's piece yield equivalent mathematics, and demand only a singular intermediate notion of infinity, I.

D < I < C

What's astonishing is that, you can prove that there are infinite subsets of the real numbers that are of size D, I, and C. But no other sizes are allowed. No explicit axioms are required.

Put differently, it could well be that our ZFC-axioms of set theory were simply not refined enough to capture these nuances of the real numbers.

I’ll leave the details to Natalie’s article, which may require more than one read to really enjoy properly.

References

A useful textbook in these matters might be Notes on Set Theory by Yiannis Moschovaskis. You might also check out Paul Cohen’s 1956 lectures. Cohen of course proved that the continuum hypothesis was independent of ZFC.

The actual proof that relates the two new axiom sets is provided in the Annals of Mathematics.

And of course, Natalie’s article in Quanta.