
06448903
j
2015d.00478
Smith, Eric D.
G\"odel's incompleteness and consistency theorems elucidated with principles of abstraction levels, complementarity, and selfreference.
Philos. Math. Educ. J. 27, 25 p., electronic only (2013).
2013
Professor Paul Ernest, University of Exeter, Graduate School of Education, Exeter
EN
E30
E20
mathematics and philosophy
G\"odel's theorems
complicated systems
complexity
emergence
concept structuring
complementarity
dualistic principle
levels of abstraction
mathematical logic
expressive systems
selfreference
fractals
paradoxes
metamathematics
arithmetic
systems engineering
SysML modelling
incompleteness
inconsistency
syntactic
semantic
axiomatization
deductive
http://people.exeter.ac.uk/PErnest/pome27/Smith%20%20Godels%20Incompleteness%20and%20Consistency%20Theorems.docx
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 selfreference. Selfreference 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. G\"odel's Incompleteness and Inconsistency Theorems prove truths about formal languages that have the ability of selfreference, 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.