Intuition versus Formalization: Some Implications of Incompleteness on Mathematical Thought

PDF Version Also Available for Download.

Description

This paper describes the tension between intuition about number theory and attempts to formalize it. I will first examine the root of the dilemma, Godel's First Incompleteness Theorem, which demonstrates that in any reasonable formalization of number theory, there will be independent statements. After proving the theorem, I consider some of its consequences on intuition, focusing on Freiling's "Dart Experiment" which is based on our usual notion of the real numbers as a line. This experiment gives an apparent refutation of the Axiom of Choice and the Continuum Hypothesis; however, it also leads to an equally apparent paradox. I conclude ... continued below

Physical Description

v, 40 leaves

Creation Information

Lindman, Phillip A. (Phillip Anthony) August 1994.

Context

This thesis is part of the collection entitled: UNT Theses and Dissertations and was provided by UNT Libraries to Digital Library, a digital repository hosted by the UNT Libraries. It has been viewed 26 times . More information about this thesis can be viewed below.

Who

People and organizations associated with either the creation of this thesis or its content.

Chair

Committee Member

Publisher

Rights Holder

For guidance see Citations, Rights, Re-Use.

  • Lindman, Phillip A. (Phillip Anthony)

Provided By

UNT Libraries

With locations on the Denton campus of the University of North Texas and one in Dallas, UNT Libraries serves the school and the community by providing access to physical and online collections; The Portal to Texas History and UNT Digital Libraries; academic research, and much, much more.

Contact Us

What

Descriptive information to help identify this thesis. Follow the links below to find similar items on the Digital Library.

Degree Information

Description

This paper describes the tension between intuition about number theory and attempts to formalize it. I will first examine the root of the dilemma, Godel's First Incompleteness Theorem, which demonstrates that in any reasonable formalization of number theory, there will be independent statements. After proving the theorem, I consider some of its consequences on intuition, focusing on Freiling's "Dart Experiment" which is based on our usual notion of the real numbers as a line. This experiment gives an apparent refutation of the Axiom of Choice and the Continuum Hypothesis; however, it also leads to an equally apparent paradox. I conclude that such paradoxes are inevitable as the formalization of mathematics takes us further from our initial intuitions.

Physical Description

v, 40 leaves

Subjects

Keywords

Library of Congress Subject Headings

Language

Identifier

Unique identifying numbers for this thesis in the Digital Library or other systems.

Collections

This thesis is part of the following collection of related materials.

UNT Theses and Dissertations

Theses and dissertations represent a wealth of scholarly and artistic content created by masters and doctoral students in the degree-seeking process. Some ETDs in this collection are restricted to use by the UNT community.

What responsibilities do I have when using this thesis?

When

Dates and time periods associated with this thesis.

Creation Date

  • August 1994

Added to The UNT Digital Library

  • March 24, 2014, 8:07 p.m.

Description Last Updated

  • April 23, 2015, 1:37 p.m.

Usage Statistics

When was this thesis last used?

Yesterday: 0
Past 30 days: 1
Total Uses: 26

Interact With This Thesis

Here are some suggestions for what to do next.

Start Reading

PDF Version Also Available for Download.

Citations, Rights, Re-Use

Lindman, Phillip A. (Phillip Anthony). Intuition versus Formalization: Some Implications of Incompleteness on Mathematical Thought, thesis, August 1994; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc277970/: accessed August 17, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .