Gödel’s incompleteness and consistency theorems elucidated with principles of abstraction levels, complementarity, and self-reference. (English)

Philos. Math. Educ. J. 27, 25 p., electronic only (2013).

Summary: The question “What is a system?" can be asked and answered differently, but the fact that the question refers to a whole ‒ called a system ‒ remains. While formal engineering design and modeling languages describe system parts, the practice of systems engineering results when there is reference to holistic systems, often via self-reference. Self-reference creates the possibility of circular, paradoxical reasoning where multiple outcomes can occur. Conceptual structuring by abstraction levels with complementarity clarifies paradoxes without resort to strict hierarchical decomposition that nullifies complexity. Gödel’s Incompleteness and Inconsistency Theorems prove truths about formal languages that have the ability of self-reference, elucidating analogous relations among: informal natural language statements about systems, systems, and formal languages that describe systems. The goal of this work is to foster cognizance in system descriptions.