This note is a part of my Zettelkasten. What is below might not be complete or accurate. It is also likely to change often.
30th October, 2021

Formal Systems

A formal system consists of just two things:

  • Axioms (or foundational theorems) - zero, one or infinitely many
  • Rules of production (or rules of inference) - A bunch of rules used to create theorems using axioms.

Using these two, we can create theorems. Theorems are states in the formal system that can be reached from axioms using rules.

An example format system could be:

  • Axiom: MI
  • Rules:
    • MxI -> MxIU
    • Mx -> Mxx
    • MxIIIy -> MxUy

Using these two things, we can make theorems like MII, MIU, etc. But x and y are not part of the system. As something outside the system, we can assign meaning to them, but inside the system they don't have any meaning.