Is there an infinite number of number sets? (natural, integers, real, complex...)

Question

I've heard of a number set on that's the smallest set after complex numbers, according to http://www.askamathematician.com/2012/09/q-is-there-a-number-set-that-is-above-complex-numbers/ (The Physicist) called quaternions.

A comment on that article says that there is another set called dual numbers, that's used to rendering fractals in graphics and even have equivalent quaternions, as stated by blackhole0173.

I wonder if there are more number sets to discover!

For each prime $p$, I would count the $p$-adic numbers $\mathbb{Z}_p$, or $\mathbb{Q}_p$, as a "number set" which makes infinitely many. Also, any number field such as $\mathbb{Q}(\sqrt{n})$ for $n \ne 1$ square-free, or its ring of integers, would count as a "number set" in my book. — Daniel Schepler, Oct 15 '18 at 17:56
you might be interested to read about the Cayley-Dickson construction. — Matthew Towers, Oct 15 '18 at 18:06

score 1 · Accepted Answer · answered Oct 15 '18 at 17:54

The problem with your question is that nobody knows what a "number set" is. The way I interpret it, each positive integer $n$ creates a "number set" of modular arithmetic mod $n$, and there are infinitely many, so yes.

If you were just thinking of fields/division rings, then there's always infinite towers of fields, e.g. infinitely many subfields of the algebraic closure of the field of two elements.

I would encourage you to do a little bit of reading here about what concepts are used in mathematics that are colloquially referred to as "numbers", so that you become more familiar with the issue.

Is there an infinite number of number sets? (natural, integers, real, complex...)

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