I just read something cool:
• Joel David Hamkins, Nonstandard models of arithmetic arise in the complex numbers, 3 March 2018.
Let me try to explain it in a simplified way. I think all cool math should be known more widely than it is. Getting this to happen requires a lot of explanations at different levels.
The Peano axioms are a nice set of axioms describing the natural numbers. But thanks to Gödel’s incompleteness theorem, these axioms can’t completely nail down the structure of the natural numbers. So, there are lots of different ‘models’ of Peano arithmetic.
These are often called ‘nonstandard’ models. If you take a model of Peano arithmetic—say, your favorite ‘standard’ model —you can get other models by throwing in extra natural numbers, larger than all the standard ones. These nonstandard models can be countable or uncountable. For more, try this:
• Nonstandard models of arithmetic, Wikipedia.
Starting with any of these models you can define integers in the usual way (as differences of natural numbers), and then rational numbers (as ratios of integers). So, there are lots of nonstandard versions of the rational numbers. Any one of these will be a field: you can add, subtract, multiply and divide your nonstandard rationals, in ways that obey all the usual basic rules.
Now for the cool part: if your nonstandard model of the natural numbers is small enough, your field of nonstandard rational numbers can be found somewhere in the standard field of complex numbers!
In other words, your nonstandard rationals are a subfield of the usual complex numbers: a subset that’s closed under addition, subtraction, multiplication and division by things that aren’t zero.
This is counterintuitive at first, because we tend to think of nonstandard models of Peano arithmetic as spooky and elusive things, while we tend to think of the complex numbers as well-understood.
However, the field of complex numbers is actually very large, and it has room for many spooky and elusive things inside it. This is well-known to experts, and we’re just seeing more evidence of that.
I said that all this works if your nonstandard model of the natural numbers is small enough. But what is “small enough”? Just the obvious thing: your nonstandard model needs to have a cardinality smaller than that of the complex numbers. So if it’s countable, that’s definitely small enough.
This fact was recently noticed by Alfred Dolich at a pub after a logic seminar at the City University of New York. The proof is very easy if you know this result: any field of characteristic zero whose cardinality is less than or equal to that of the continuum is isomorphic to some subfield of the complex numbers. So, unsurprisingly, it turned out to have been repeatedly discovered before.
And the result I just mentioned follows from this: any two algebraically closed fields of characteristic zero that have the same uncountable cardinality must be isomorphic. So, say someone hands you a field F of characteristic zero whose cardinality is smaller than that of the continuum. You can take its algebraic closure by throwing in roots to all polynomials, and its cardinality won’t get bigger. Then you can throw in even more elements, if necessary, to get a field whose cardinality is that of the continuum. The resulting field must be isomorphic to the complex numbers. So, F is isomorphic to a subfield of the complex numbers.
To round this off, I should say a bit about why nonstandard models of Peano arithmetic are considered spooky and elusive. Tennenbaum’s theorem says that for any countable non-standard model of Peano arithmetic there is no way to code the elements of the model as standard natural numbers such that either the addition or multiplication operation of the model is a computable function on the codes.
We can, however, say some things about what these countable nonstandard models are like as ordered sets. They can be linearly ordered in a way compatible with addition and multiplication. And then they consist of one copy of the standard natural numbers, followed by a lot of copies of the standard integers, which are packed together in a dense way: that is, for any two distinct copies, there’s another distinct copy between them. Furthermore, for any of these copies, there’s another copy before it, and another after it.
I should also say what’s good about algebraically closed fields of characteristic zero: they are uncountably categorical. In other words, any two models of the axioms for an algebraically closed field with the same cardinality must be isomorphic. (This is not true for the countable models: it’s easy to find different countable algebraically closed fields of characteristic zero. They are not spooky and elusive.)
So, any algebraically closed field whose cardinality is that of the continuum is isomorphic to the complex numbers!
For more on the logic of complex numbers, written at about the same level as this, try this post of mine:
• The logic of real and complex numbers, Azimuth 8 September 2014.