Category Archives: Formal Systems

In this post we will introduce a concept invented by Gödel called Gödel numbering.  Essentially, Gödel numbering allows one to look at formal systems in a different way – a way that allows one to interpret the system within mathematics.  We will do … Continue reading →

We are now going to make explicit what the pq-System was designed to model; although, you may have already guessed it.  In short, the pq-System is designed to model the addition of two natural numbers.1 Here is the translation for xpyqz:   … Continue reading →

We will now expand our typographical T function1 for the pq-System by incorporating the rule of inference, which is as follows…    Rule of Inference: If x, y, and z all stand for particular strings made up of hyphens only, and if xpyqz … Continue reading →

In our last post we considered the deductive apparatus of the pq-System.  In this post we will begin to consider the motivation for this system, i.e., what the system is intended to model.   pq-System Revisited Alphabet: p q – Axiom Schema: xp–qx– is … Continue reading →

In the last post the langauge of the pq-System was introduced.  Here is a summary… Alphabet: p q – Expressions of the pq-System:  E is an expression of the pq-System if and only if E is made up of a finite number of symbols … Continue reading →

Raymond Smullyan’s  Gödel’s Incompleteness Theorem begins with an abstract version of Gödel’s First Incompleteness Theorem.  We will soon look at that proof, but before we do that we need to explain what a formal system is because the objects of Gödel’s argument are certain types of … Continue reading →