Gödel's Incompleteness Theorem

A formal system (one that is consistent never yields a false statement) can not also be a complete system (containing all true statements)–there will always be statements that are unprovable yet true (i.e. G-statement).

The proof is derived from Principia Mathematica (PM) logic looking at number theory. You can build new theorems from axioms that are true infinitely but, because it is infinite, it can not contain every true statement.

That means there is always some truths that can’t be captured by a single system no matter how elaborate or seemingly robust. That’s why it’s important to have multiple systems and acknowledge you will always be limited in understanding the fullness of anything described by a formal system.

See also:

• I Am a Strange Loop provides a nice explanation about recursive thinking that led Godel to finding a proof
• Gödel Incompleteness For Startups applies this theorem to building companies and competition

Links to this note

• G-Statement

  An unprovable, but true statement as described in Gödel’s incompleteness theorem. Any formal system will be incapable of capturing all true statements and so there will always be unprovable, but true statements about which the system tries to describe.

• Cynicism Is the Opposite of Optimism and Pessimism

  Cynics believe that nothing ever changes at a fundamental level. By it’s nature, cynicism is a dead end where everything stays the same over time.

• Experience Is an Illusion of System Completeness

  We tend to think of experience in a given role is obviously good–it provides pattern matching and intuitions built up on lived events. If you consider experience as a process of formalization (i.e. formal system) of a given profession, then Godel incompleteness means there will always be more unproven yet true axioms in the field.

• Formal System

  A formal system consists of axioms (statements that are true) and rules (how axioms can be manipulated). New axioms can be deduced from other axioms and rules. You formalize a system when you start defining these statements and rules.

• Gödel Incompleteness for Startups

  An essay that relates Gödel’s incompleteness theorem (along with the Halting Problem) to startup disruption—arguing that all successful startups discover one or more G-statements and extract value by building a formal system around it.

• Tupper’s Self-Referential Formula Is a Visual Loop

  Tupper’s formula displays its formula when plotted on a graph. This is a form of recursion but more like a mirror. Compared to Godel’s incompleteness theorem (also self-referential), Tupper’s formula is more like perception not reception—it does not create new meaning in a system unintended to do so (Principia Mathematica) and is more like a reflection of itself.

• Münchhausen Trilemma Explains Common Tropes of Arguments

  The Münchhausen trilemma occurs when attempting to prove anything to be true. Such attempts fall into three tropes—a circular argument which supports itself (A <-> B), a regressive argument where the proof requires further proof infinitely (“why?” x infinity), and a dogmatic argument which relies on an assertion which is not defended (“because”).

• Creating Meaning Through Analogy

  One of the most profound findings from Godel’s incompleteness theorem is that meaning can be mapped onto a system that was specifically designed to prevent it. Principia Mathematica, the system Godel was poking at, was designed—amongst other reasons—to solve the paradoxes of set theory and logic with a constrained set of rules. Godel abstracted over the symbolic logic (converting proofs into numbers) to show a self-referencing statement could not only be represented in PM but was also undecidable.

• Godel’s Loophole Is a Constitutional Flaw That Could Enable a Legal Dictatorship

  While the exact details are unknown, Godel claimed to have discovered a contradiction in the US Constitution that could allow the democracy to be legally turned into a dictatorship. There is much speculation about what the flaw could be and according to some accounts he told Einstein who had to calm him down.

• A Strange Loop Gives Rise to Human Selfhood

  A strange loop is a combination of traits that creates the condition for selfhood. Categories of numerous symbols derive meaning from raw stimuli. Categorization leads to perception rather than reception. Abstractions create reality and high-level behavior no longer consists of lower-level behavior only. The loop reinforces the idea of the ‘self’ and the self feels the most real.

• The Downside of First-Principles Thinking

  The problem with first-principles thinking is that you don’t know what you don’t know. You might end up deriving the same thing you could have read in a 101-level textbook. You might think you have a novel idea of a blockchain-powered utopia but it turns out it’s Georgism with extra steps.