Gödel’s Incompleteness Theorem Applied to the Software Industry

This document is specially designed for an audience with little preparation in mathematics, thus, the concepts involved have been synthesized so that they can be easily digested.


In recent years, considering factors such as economies of scale, business competitiveness, and rapid deployment of technologies, the demand for software solutions in record times has been increasing. 

Nowadays, it is necessary to consider not only the software development but the implementation of quality control strategies that avoid errors in the final solutions. 

From the above, a valid question emerges that represents the cornerstone of this document: When is a result considered acceptable or how is it guaranteed that the solution obtained is the best that can be achieved under certain initial conditions? Does the perfect system exist?

In order to provide a solid answer, we use a mathematical argument: Gödel’s Incompleteness Theorem (also known as Gödel’s Theorem).


First, we clarify some definitions.

Theorem: A statement that can be proved true by mathematical operations and logical arguments. Therefore, a theorem has already been confirmed and validated, sometimes it even needs the approval of committees and scientists of great renown. 

Now, we explain the concepts of system and Aristotle’s true knowledge process, so we use Figure 1:  

Figure 1. Explanation on system, phenomenon and Aristotle’s true knowledge process


Then the description goes as it follows:

We encounter phenomena (P) or events, which we want either unconsciously or consciously, to understand them and then control or predict them. In this way, a system (S), which is a set of explanations or rules (R), is generated to try to characterize, model, or replicate the phenomenon. Aristotle’s process of true knowledge refers to the general function that unites both the system and the phenomenon. 

Just to mention one example, we have the Constitution (or laws in general); according to the official definition, it is a set of rules (system) that regulate the phenomenon of how a state or country should be governed. 

The list of cases is more extensive, nevertheless we have listed the most representative one. 


Published in 1931 by the philosopher and mathematician Kurt Gödel, it states that if we take any system S that describes a phenomenon P, we will always find features or elements in P (called true statements) that can NOT be explained using S

Figure 2 helps us understand this visually:

Figure 2. Gödel’s Theorem graphical representation


This translates as the perfect system does not exist and this applies to any system, regardless of industry, geography, belief or language. 


To structure these effects, they are divided into two categories: qualitative and quantitative.

On the qualitative ones, we conclude that some processes will never stop being carried out as there will always be imperfections and they must be assigned the corresponding resource. An example is that of Cybersecurity, in the areas where they are dedicated to fix bugs as in Microsoft; since the scenario in which all vulnerabilities are resolved will never be reached, at least more efficient strategies are developed to deal with the problem. 

From the quantitative standpoint, we can count on metrics and techniques that help us generate the best possible solution, for example, in Data Science there is a method called ensemble modeling in which two or more analytical techniques are combined to generate a more robust one that contains all the strengths of the previous ones. 

Another quantitative example occurs in the processes that provide sections dedicated to mitigating the unknown or risk, as precisely does the area of Risk Management, SCRUM methodology or Statistics which has formulas to give an approximation in those cases. 


From all the existing systems, there is one that stands out and that concerns us as a species, to explain this, we take the Figure 3:


Figure 3. The human being is a system too.

As the image shows, the human being is a system at both singular and plural (society) levels that adheres to the same effects shown in Gödel’s theorem, therefore, the human being is also incomplete and imperfect. 

With this result we argue that everyone is a system that will never be able to describe reality in its entirety, but by uniting our systems as a group, we can achieve something that at the individual level would not have been possible. 


Through this article we have confirmed that there is no such thing as the perfect system and instead of being a disappointment or a negative aspect, this helps us to develop, among other things, a correct orientation of resources to generate quality results and at an introspective level, empathy, tolerance and a sincere collaboration between human beings.