Are mathematics a reliable way to explain reality? Can we trust them? And what role could Python play in such questions?
Gödel's Incompleteness Theorems sit at the foundation of mathematics and computer science: they show that there are inherent limits to how far we can formalize truth. They are also famously hard to approach. In this 30-minute talk at EuroPython 2026 in Kraków, I used Python not as a language to optimize, but as a conceptual tool — a way to make Gödel's First Incompleteness Theorem tangible through algorithmic thinking.
The talk walks from Hilbert's dream of a complete, decidable mathematics, through Turing machines and the Church–Turing thesis, to the halting problem and a short self-referential program. The same diagonal trick, in a different costume, is then aimed at formal systems themselves. The result is the incompleteness theorem: any logical system that is sound and expressive enough to talk about programs contains true statements that cannot be proven.
The ideas are developed at full length in my bachelor's thesis and in the blog series that expands the same argument with structured proofs.
Key themes covered
- Hilbert's programme, and why completeness and decidability failed
- Python programs as mathematical objects, via the Church–Turing thesis
- Decidability, semi-decidability, and the halting problem
- Formal systems, soundness, completeness, and the gap between proof and truth
- Gödel's First Incompleteness Theorem, reached through a self-referential Python program
