Gödel’s Incompleteness Theorems: History, Proofs

Gödel’s Incompleteness Theorems: History, Proofs, Implications

In 1931, a 25-year-old Kurt Gödel published a paper in mathematical logic titled “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.” This paper contained the proofs of two remarkable “incompleteness theorems,” which state:

For any consistent axiomatic formal system that can express facts about basic arithmetic,

1. there are true statements that are unprovable within the system
2. the system’s consistency cannot be proven within the system

Gödel’s theorems were immediately recognized as a pivotal achievement in logic and the foundations of mathematics. Since then, the incompleteness theorems are have been referenced in everything from philosophy to popular science but they are seldom carefully examined. Within philosophy of mathematics, Gödel struck a decisive blow to the logicist and formalist programs of Frege, Russell, and Hilbert–the effort, going back to Frege’s work in the late 1800s and carried forward by Whitehead and Russell’s Principia Mathematica and Hilbert’s formalist philosophy of mathematics, to provide a provably consistent axiomatic foundation for all of mathematics. Gödel showed that no formal (i.e., syntactic) system is sufficient to capture the truths of even arithmetic–leave alone all of mathematics. We will discuss this historical and philosophical context before delving into the proofs of the incompleteness theorems. The level of mathematical detail will be calibrated according to the interests and abilities of the class.

We will work with propositional and predicate logic; formal axiomatic systems and formal proofs (and the consistency and completeness of such); Peano arithmetic; the arithmetization of syntax via Gödel numbering; effective/computable enumerability (via computable functions and Turing machines); and finally the famous Gödel sentence: the diagonalization argument that leads to a self-referential formula of arithmetic which says “I am not provable” (a syntactic version of the liar paradox). Finally, we will discuss the influence and impact of Gödel’s theorems on mathematics and the philosophy of mathematics, as well as the “uses and abuses” of Gödel incompleteness in fields such as cognitive science, philosophy of mind, artificial intelligence, and even political science and sociology.

Readings will include selections from Rebecca Goldstein’s Incompleteness: The Proof and Paradox of Kurt Gödel, Peter Smith’s An Introduction to Gödel’s Theorems, Douglas Hofstadter’s Gödel, Escher, Bach; and John Dawson’s Logical Dilemmas: The Life and Work of Kurt Gödel.