A Simple Explanation

Here’s a good test of skill — make a very complex idea seem pretty simple. In just two paragraphs, I explain one hifalutin math theorem, a topic usually considered in graduate-level courses in number theory.

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Gödel’s First Incompleteness Theorem: To every w-consistent recursive class k of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(k) (where v is the free variable of r).

Suppose you have a system of logic as sophisticated as grade-school arithmetic. You have numbers, operations (like addition and subtraction), and relations (such as 2 is less than 3). You have true statements (the sum of two even numbers is even) and you have false statements (the sum of two odd numbers is odd). What Gödel’s theorem tells us is that some statements are undecidable; i.e., there’s no way to determine whether they’re true or false by using the rules and axioms of grade-school arithmetic.

Gödel’s theorem has BIG implications. It insures that, in any complex system of logic, there will be statements that cannot be proven true or false by using the rules and axioms of that system. It might be possible to go outside the system to determine whether some statement is true or false, but that just compounds the problem since the larger, more complex system will also yield undecidable statements, and so on. And none of this is a matter of our limited intellectual abilities. It’s in the nature of things. Gödel’s theorem insures that even the gods can’t reckon everything — not logically.