Before Gödel, was undecidability of axiomatic systems an issue at all?

Question

Before Gödel, was the issue raised that there may be undecidable statements within axiomatic systems of thought?

Gödel managed to answer affirmatively by proving that the assumption of the concinstency of axioms of arithmetic is not only a meta-statement, but also a statement within arithmetic, indeed an undecidable one.

So in the long tradition of philosophy & mathematics from antiquity to the present, did someone even remotely suspect that undecidable statements might exist at least in principle, or was decidability considered to be self-evident for axiomatic systems?

Answer 1

Godel himself said that all he did was formalise the Cretan liar paradox into a formal system. So the idea or notion of undecidable statements was already apparent a long time before Godel but obviously not phrased in such terms.

A simple parallel is with arithmetic. Its easy enough to notice that putting things into a group has certain properties which is more or less obvious to anyone, but is formalised into what we call addition.

Answer 2

While it’s not quite a perfect parallel, a related concern about decision problems had been phrased some years before Gödel’s work: see https://en.m.wikipedia.org/wiki/Entscheidungsproblem . David Hilbert was pretty convinced as to the decidability of arithmetic prior to the Incompleteness/Incomputability proofs, but was aware that the problem remained open.

Answer 3

Before Frege, axiomatic systems were not a focus of philosophy, and Goedel is pursuing the immediate upshot of Frege's failure. So, in some sense, no. Nobody cared. Mathematics was grounded in some internal, perfect mental reality and not really based on axioms. Axioms just helped keep things clear.

Paradoxes abound throughout the history of philosophy. But they were often addressed ad-hoc and blamed on inadequate clarity. Frege made a real attempt to describe all use of language in axiomatic terms, hoping a mathematical method could ground these issues.

Instead, it imported the problems of language back into mathematics. It prompted Russell and others to point out that these problems infect even the simplest intuitive notions that make us accept math. After that, it became clear that the notion of a consistent Platonic model of math was inadequate.

Hilbert realized that adopting an explicit ontology for math, like Brouwer did with Kantian notions, generally did not accord with the overall way math works. It made different parts of math either fall apart or become question-begging morasses, full of effectively circular arguments.

He wanted the reliability and scope of mathematics to be grounded entirely in symbolic manipulations. The approach to logic through models embedded entirely in sets of symbols, not referring in any way to any outside domain of reference, seemed like an answer. This is the point at which axioms then become the real basis of math, which they never were even for Euclid.

The existence of undecidable statements answers Hilbert in the negative -- this approach to mathematics cannot remain adequately simple, prove the consistency of mathematics, and still describe the whole of mathematics. Mathematics is not entirely determined by its process, it needs some source of external contents to decide the 'undecidable truths'.

That means that mathematics cannot save logic from its problems. And therefore logic cannot be a domain of its own divorced from external grounding.