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Journal of Undergraduate Research

Keywords

teaching incompleteness, finite set theory, informal English, paradoxes

College

Humanities

Department

Philosophy

Abstract

In the 1930s, a 24-year-old mathematician named Kurt Gödel developed a theorem showing that certain mathematical and logical systems, those capable of defining numbers, arithmetic, and multiplication, must be incomplete. These systems are incomplete, he showed, because they can generate certain true, but unprovable, statements. Gödel’s theorem has since become a common concept taught in advanced undergraduate logic courses. Undergraduate philosophy students attempting to understand incompleteness are usually well-grounded in basic Aristotelian and first-order predicate logic, as well as set theory. When they try to tackle Gödel’s proof, however, they are thrown into a dizzyingly unfamiliar new world of notation and number coding. The purpose of this project was to develop a new method of proving incompleteness in finite set theory, a language that is familiar to students who have completed an average intermediate logic course. By leading students through the incompleteness theorem using finite set theory as both the object language and metalanguage, this method helps students more quickly and thoroughly grasp the concept of incompleteness.

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