Mandraveli, Chrysanthi Aikaterini (2025) Teaching Incompleteness of Arithmetic and the Paris-Harrington Theorem. Bachelor's Thesis, Computing Science.
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Abstract
The Incompleteness of Arithmetic refers to the surprising phenomenon that not all true statements about the natural numbers can be proven within a formal system such as Peano Arithmetic. This result was first discovered by Kurt Gödel in 1931, who showed that every sufficiently powerful and consistent formal system contains true statements that it cannot prove. This project investigates this phenomenon through a detailed study of the Paris-Harrington Theorem (1977), a mathematically natural and combinatorial demonstration of the incompleteness phenomenon. The Paris-Harrington theorem illustrates that certain true mathematical statements cannot be proven within standard arithmetic systems such as Peano Arithmetic, a widely used formal system that encodes the properties and operations of the natural numbers. Unlike Gödel’s original examples, which rely on intricate self-referential sentences (i.e. statements that speak about their own provability), the Paris-Harrington theorem utilized Ramsey Theory, a branch of combinatorics concerned with conditions under which structure necessarily appears in large, seemingly random arrangements. The thesis aims to make this deep logical result approachable for advanced undergraduate students in Mathematics, Computer Science and AI at the Bernoulli Institute, University of Groningen by developing an eight-week course supported by carefully designed teaching materials, with this reader as its central resource.
| Item Type: | Thesis (Bachelor's Thesis) |
|---|---|
| Supervisor name: | Ramanayake, D.R.S. and Akbar Tabatabai, S.A. |
| Degree programme: | Computing Science |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 17 Jul 2025 10:53 |
| Last Modified: | 17 Jul 2025 10:53 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/36104 |
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