Gödel's Incompleteness Theorem and Numerical Analysis

Question

I am no expert in Mathematical Logic so I can't really express my question formally but I do hope that it will make sense.

As far as I know the implication of Gödel's incompleteness theorem for Mathematics is that we can't prove every aspect of Mathematics. In order to keep moving forward proving more theorems we need more axioms. So this procedure will keep happening infinitely and thus Mathematics are proven to be an inexhaustible subject.

What I keep thinking is does Gödel's incompleteness theorem apply to a field like Numerical Analysis?

My thoughts are that Numerical Analysis focuses on finding efficient numerical solutions to problems and not so much on axioms, proving theorems and building theories. Of course those exist but it is not the primary concern to build and prove elaborate theories that have no or little practical use. In Euclidean Geometry for instance everything is build upon axioms so one can add axioms and prove theorems about angles and triangles etc. forever no matter if the implications of those theorems are important or not. Can something like that be said about Numerical Analysis?

If there is something wrong enlightens me because, as I mentioned, I am not very familiar with Mathematical Logic.

Answer 1

Like any field of mathematics, Numerical Analysis certainly does have theorems, and Gödel's results apply to them.

If you insist on restricting your attention to applications involving computations on a specific machine with a finite amount of memory, then there is no scope for incompleteness: there are only finitely many programs that fit in the memory, and the state space of the computer is finite; any program can be simulated until it either terminates or provably enters an infinite loop.

But that's a very narrow focus. Numerical analysts usually work in a model that treats the available memory as infinite. Then the unsolvability of the Halting Problem, which is very closely related to Gödel's incompleteness results, is very relevant.

Here's a specific example (Richardson's theorem). Suppose you have an expression $A(x)$ involving rational numbers, the number $\pi$, the number $\log 2$, the variable $x$, the operations of addition, subtraction, multiplication, composition, and the functions $\sin$ and $\exp$. You would like to determine whether there is some real number $x$ such that $A(x) < 0$. Richardson's theorem says there is no algorithm that can do this.

Answer 2

Mathematics is indeed moving forward - the more we understand the more new questions suggest themselves. Some of those questions touch on practical applications, some are purely theoretical (for now, at least). But only a small part of what's new is connected to formal logic. We don't need more axioms to prove more theorems, we need to understand the implications of what we have better.

I don't know of any part of numerical analysis that's affected by new (or even old) results in logic.