Gödel’s Incompleteness Theorems: True Statements That Cannot Be Proved

We are approaching the midpoint of our experiment — each new post is shaped by your comments on the previous day’s entry. Write more, tomorrow’s topic might be yours! Yesterday, one comment stood out:

“Well, after the Gödel incompleteness theorem appeared in the 1940s, there was a real chance that Fermat’s Last Theorem could have simply remained an unprovable truth.”

…and that’s fair: the concern was genuine. In the 1960s–1980s, following the work of Kurt Gödel (1931) and especially Paul Cohen’s method of forcing (1963), it became clear that there exist true statements that cannot be proved within standard axiomatic systems. Against this backdrop, a number of mathematicians seriously entertained the idea that Fermat’s Last Theorem (FLT) might be independent of the accepted axioms. This anxiety surfaced in the press as well: in the late 1980s, on the heels of several failed “breakthroughs,” articles appeared asking “what if FLT is fundamentally unprovable?”

By 1995, Andrew Wiles put those doubts to rest: a proof of FLT was found — via the modularity of semistable elliptic curves. Or… was it?

What the Gödel Incompleteness Theorems Actually Say

First theorem (in plain terms): in any consistent formal system capable of expressing ordinary arithmetic of natural numbers, there will always exist a true statement that can neither be proved nor disproved within that system. This is called independence: to reach the truth, you need stronger axioms.

Second theorem: such a system cannot prove its own consistency (assuming it is, in fact, consistent). In other words, it is impossible to “close off mathematics once and for all” with a single fixed set of rules — there will always be a door leading outside.

This was a shock. It turned out that not everything can be proved within a single “mathematics.” Sometimes new axioms must be added. As a result, mathematicians began compiling lists of such unprovable statements.

Classic Examples of Gödel Incompleteness in Action

The Continuum Hypothesis (CH): does there exist a cardinality strictly between the set of integers and the set of real numbers? Within the standard Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC), it can neither be proved nor disproved: Gödel showed CH is consistent with ZFC, and Cohen showed that its negation is consistent with ZFC as well.

The Axiom of Choice (AC) relative to ZF (Zermelo–Fraenkel without the Axiom of Choice) — neither AC nor its negation can be derived from ZF alone.

Whitehead’s problem in abelian group theory and Suslin’s hypothesis about ordered sets — both are undecidable within ZFC.

Goodstein’s theorem: every “Goodstein sequence” eventually reaches zero. This is true, but cannot be proved within Peano Arithmetic (PA).

Kruskal’s tree theorem and Harvey Friedman’s results: an entire class of natural combinatorial statements is only resolvable by adding large cardinals — that is, they lie beyond ZFC.

Undecidable vs. Unprovable: An Important Distinction

An algorithmically undecidable problem is about computability: there is no program that can answer “yes/no” for all possible inputs (the classic example being the Turing halting problem). Unprovable in Gödel’s sense is about axioms: you may have a specific true statement, but your chosen axioms are simply not strong enough to prove it — add a new axiom, and the proof becomes possible.

Independence — the inability to either prove or disprove something — is a property relative to an axiomatic system, not an absolute and eternal verdict.

Does the Gödel Incompleteness Theorem Apply to Fermat’s Last Theorem?

Returning to the proof of FLT: the question of its “provability” remains genuinely open. Wiles proved FLT by relying on deep objects from set theory and geometry — modular forms, Galois representations, deformation theory, and more — all of which reach far beyond the confines of Peano Arithmetic. We still do not know whether Fermat’s Last Theorem follows from the standard axioms of arithmetic.

It is entirely possible that while the theorem is provable in ZFC, it may not be provable in the weaker system of Peano Arithmetic. If this were ever demonstrated, FLT would join the ranks of statements like Goodstein’s theorem — natural assertions about natural numbers that are true, yet unprovable in PA.