The Infinity Conundrum: Understanding Topics in Set Theory and the Continuum Hypothesis

Sabrina Helck

Name: Sabrina Helck
Major: Mathematics
Advisors: Dr. Subhadip Chowdhury; Dr. Rob Kelvey (second reader)

This project is concerned with articulating the necessary background in order to understand the famous result of the undecidability of the continuum hypothesis. Throughout this thesis, I discussed the necessary background needed to accomplish this. The first portion of this independent study discusses the foundations of set theory, stating fundamental definitions and theorems that will be used throughout the remainder of the project. The second portion focuses on ordinal and cardinal numbers which will directly relate to the final section. Next, we discuss Zermelo-Fraenkel set theory and the axiom of choice (ZFC) which introduces the idea of set theoretic systems and models. The final section states the continuum hypothesis, as well as the weak continuum hypothesis and the generalized continuum hypothesis. The famous result of the continuum hypothesis is that it is unprovable in ZFC. In other words, it is independent of the system. One mathematician, Kurt Gödel, proved that the continuum hypothesis cannot be proven false in ZFC, and later another mathematician, Paul Cohen proved that it cannot be proven true in ZFC. This thesis is only concerned with understanding Gödel’s proof. By the end of this project, we will have discussed sufficient background and language to make sense of the proof.

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Posted in Comments Enabled, Independent Study, Symposium 2022.

4 responses to “The Infinity Conundrum: Understanding Topics in Set Theory and the Continuum Hypothesis”

  1. Mary Helck says:

    Were you surprised to discover that the continuum hypothesis is unprovable in ZFC? Great job on your IS!!!

  2. Chan Sok Park says:

    Sabrina, congratulations on this important achievement! Best wishes for your post-graduation plan!

  3. Carina Arnosti says:

    Congrats Sab!

  4. Sabrina Helck says:

    Thank you! The fact that it was unprovable was the thing that intrigued me to pursue this project, but it was very interesting to see how mathematicians arrived at the result.