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Solving the undecidability of the continuum hypothesis: a short summary of the results since 1963

Kuijer, L.B. (2007) Solving the undecidability of the continuum hypothesis: a short summary of the results since 1963. Bachelor's Thesis, Mathematics.

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Abstract

In 1891 Georg Cantor proved that there exist multiple size of infinity. In particular, the size of the natural numbers, @0, is not the same as that of the reals. This of course begs the question: is there a set larger than the natural numbers, but smaller than the reals? The Continuum Hypothesis (CH) is the statement that there is no such set. Since the reals can be seen as the powerset of the natural numbers, CH can be written as ’2@0 = @1’. Cantor tried to prove CH, but he failed (obviously, as would later be shown). CH then made it on the list of Hilbert’s 23 problems (problem 1). A first step to a solution for CH came in 1940, by Kurt G¨odel, who proved that CH is consistent with the axioms of Zermelo-Fraenkel set theory and the axiom of choice (ZFC) [4]. At that point however, it was already clear that the best solution was probably ¬CH. In 1947 G¨odel published an article where he concluded that ¬CH is probably consistent with ZFC too, and an important task in set theory would be to find an extra axiom that would decide the problem in favor of ¬CH [5]. The first part of G¨odels prediction came true in 1963 when Paul Cohen proved the consistency of ¬CH with ZFC [2]. The second part of G¨odels prediction is being worked on. I’ll discuss part of Cohen’s method to prove the consistency of ¬CH with ZFC, and some of the results in finding a new axiom that would solve the continuum hypothesis. This article is based mostly on the Bourbaki Lecture ’Progr`es r´ecents sur l’hypoth`ese du continu [d’apres Woodin]’ by Patrick Dehornoy in 2003 [3], in which results by Hugh Woodin on solving the continuum hypothesis are discussed.

Item Type: Thesis (Bachelor's Thesis)
Supervisor:
Supervisor nameSupervisor E mail
xx, xxUNSPECIFIED
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 07:28
Last Modified: 17 Apr 2019 12:18
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/8400

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