Date of Award
2018
Document Type
Thesis
Degree Name
Bachelors
Department
Natural Sciences
First Advisor
McDonald, Patrick
Area of Concentration
Mathematics
Abstract
We draw an analogy between the Turing machine model of computation and the first-order theory of Peano arithmetic. Well-formed formulae with one free variable (x) are compared to Turing machines which take natural numbers n as input. The accept, reject, and never halts outputs are compared respectively to the provability, refutability, and undecidability of (n). A computable encoding of Turing machines is compared to a Godel numbering of formulae. The existence of an undecidable proposition is compared to the non-halting behavior of a certain diagonal Turing machine when processing its own encoding. We prove the existence of such an undecidable proposition by proving Peano arithmetic weakly represents a corresponding diagonal predicate. Fundamental differences between the diagonal reasoning we employ and Godel's original proof of incompleteness are discussed through the lens of the category theoretic Lawvere's Fixed Point Theorem.
Recommended Citation
Avril, Brenton, "A PROOF OF GODEL'S FIRST INCOMPLETENESS THEOREM ADAPTED FROM THE CLASSIC PROOF OF THE UNDECIDABILITY OF THE LANGUAGE OF ANY UNIVERSAL TURING MACHINE" (2018). Theses & ETDs. 5468.
https://digitalcommons.ncf.edu/theses_etds/5468