undecim wrote:Due to Gödel's incompleteness theorem, any sufficiently powerful formal system of numbers (e.g. anything that includes the integers) has equations that cannot be solved. So if you define "number" as the solution to an equation, then you can always have more types of numbers.

One needs to be careful here.

For instance, the theory of real closed fields is complete, yet the reals obviously include the integers. Similarly for the theory of algebraically closed fields of given characteristic.

The difference here is that Gödel really does not refer to the integers, but their first-order theory (as given e.g. by Peano). And there's a few things that can complete the theory of integers in some sense (for instance, if we go beyond recursively enumerable axioms - the theory of "all statements true in the natural numbers" is complete). Things also look different if we go beyond first-order logic. Now, second-order logic has other issues (lack of a proof system), but there are a lot of variations such as second-order arithmetic.

You could also take the model-theoretic perspective and consider that, via Löwenheim-Skolem, any first-order theory that has a model at all will have models of any cardinality. But which cardinalities - and thus, in a certain sense, which "numbers" or, perhaps better, "objects" - there are is defined by the set-theoretic universe in which we operate. In that sense, Gödel cannot really be relevant to the question, as we already need to "define" what numbers we have before we can check what results we get from Gödel.

And for day-to-day mathematics, we freely speak about (and quantify over) the power set of natural numbers, so we technically operate in some form of second order logic already.

So, ultimately, Gödel is not really relevant here at all. If you go down that path and really deduce some "new numbers" from there, you need to be clear about the underlying systems and universe you're discussing.