#### Title

#### Date of Completion

5-2004

#### Degree Type

Honors Thesis - Campus Access

#### Discipline

Mathematics (MATH)

#### First Advisor

Michael C. Berg

#### Abstract

Cantor's set theory, which had already been the subject of some commotion, was the focus of even more unrest and dissatisfaction in mathematical and philosophical circles when Bertrand Russell and others pointed out certain paradoxes. People began to question the consistency of mathematical set theory. This caused David Hilbert to suggest that we should set out to prove that mathematics can be given a solid foundation, destroying all paradoxes. He suggested this should be done by embedding mathematics in symbolic logic. He wanted a proof of the existence of the Entscheidungsproblem - a machine that will decide the truth or falsity of any symbolic-logic theorem/statement.

This dream was destroyed by Kurt Gödel, who proved in a radical style that any reasonable axiomatic mathematical system contains true but unprovable assertions. Accordingly mathematics is "incomplete". Hilbert was also proven wrong independently by Alan Turing, who specifically settled Hilbert's Entscheidungsproblem. He showed that such a universal machine that Hilbert asked for does not exist.

My goal is to compare these solutions by Godel and Turing and fit them into a larger logical framework. They each tackle the Entscheidungsproblem in slightly different ways, but their methods are, in some parts, strikingly similar. My purpose is to provide an exposition on both theories and draw attention to these similarities and differences.

My analysis of Turing is drawn directly and almost entirely from his original paper, "On Computable Numbers, with an Application to the Entscheidungsproblem." My analysis of Godel, while done with reference to his own work, draws heavily from the work of Ernest Nagel and James R. Newman - both their book *Gödel's Proof* published in 1958 and (a more concise version of the same book) their essay "Goedel's Proof', printed in Volume 3 of *The World of Mathematics* in 1956.

#### Recommended Citation

Scott, Edward, "Undecidable" (2004). *Honors Thesis*. 429.

https://digitalcommons.lmu.edu/honors-thesis/429