Orderings in the Complex Plane

It is commonly known that there is not an ordering on the complex numbers \mathbb{C}, however it always struck me as strange that we simply live with that. While reading one of my textbooks in Model Theory, I came across an application of the Upward Skolem-Löwenhein Theorem which states that each infinite set can be ordered. The proof is nice and helps demonstrate just how useful the Upward Skolem-Löwenhein Theorem is. It did leave me thinking back to my Complex Variables course and if there might be a sensible way to order complex numbers.

There is a somewhat obvious way to put an order on \mathbb{C} and that is as follows: a + bi \leq c +di if b < d or if b = d, then a \leq b. This preserves the ordinary ordering of the real line, and extends it in a way such that if a complex number \alpha has imaginary part less than \beta then \alpha \leq \beta. This ordering seems nice but it does have its quirks, for example any real number is less than the number i. Not just that, but x \leq -x + \epsilon i, for all real \epsilon > 0 which doesn’t seem natural. There is also the issue that we chose the order to give “priority” to numbers of greater imaginary magnitude when this is only a choice to preserve the natural ordering of the real numbers. We could change it so that each vertical line in the complex plane is less than any line to the right of it and greater than any line to its left. We could even pick a line which is not vertical or horizontal such as straight lines that have a gradient of \frac{\pi}{4} and say that lines which are “lower” than a given line are less than it. However why stop at straight lines? It seems possible to parameterise some curves in the complex plane and apply a similar logic to them to come up with an ordering.

Another idea that I had would be to somehow parameterise a Hilbert Curve in the complex plane. This space filling curve could assign a unique real number to every complex number, and we could use that to determine if a complex number is less than another. In practice I doubt this would be useful or even feasible. The ordering it gives doesn’t really related to any intuitive sense of what numbers are less than others apart from that they would be “drawn” sooner than others.

Anyway, enough with my procrastination, I have exams to revise for.

JL

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