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.

A formal system is consistent when you can’t simultaneously prove and disprove an axiom. An inconsistent system would not be very useful e.g. a weather forecasting system that predicts it will rain and not rain at the same time.

A formal system is a complete system when all statements in the system can be proved or disproved thereby allowing you to know everything about the system.

See also:

• Gödel’s incompleteness theorem which says that a formal system can not also be a complete system.

Links to this note

• 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.

• 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.

• Knowledge Capture Loops Make for Good Systems

  Real world systems for operating a complicated process don’t start out perfectly designed complete systems. New information reveals itself only after you’ve done it a few times. Failure modes you weren’t aware of become apparent only after the system breaks.

• 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).