Introduction
As a couple of people has pointed out that the conclusion I have drawn from my last post On Science and Logic, shares some similarity to the Gödel’s Incompleteness Theorems, I have learn about it this week for the first time. It is only the 2nd week of the new year, but is certainly the most fascinating thing that I have learned this year!
Some of the information that I am about to present in this post is taken from Undefined Behavior‘s Youtube channel, linked at the end. In those videos, Undefined Behavior has concisely and clearly presented the concept and a scratch proof based on the halting problem. If the reader wish to learn more about the details and proofs of Gödel’s Incompleteness Theorems, I recommend his videos.
In this post, I will skip the proofs of the theorems, as there are already great existing sources on this subject. I will take Gödel’s Incompleteness Theorems as proven true from here on.
In this post, I will use concepts such as the formal system. If the reader wants an introduction to those concepts, here is a link to my previous post: What Are Arguments and How to Make Them.
From the very beginning of human history, we have accepted our ignorance, that there are much knowledge in the universe we have not yet acquired. But many holds the belief (and many still) that with the progression of scientific discovery, we can ultimately have an answer to every answerable question. That is, until Kurt Gödel and his 1931 paper.
Gödel’s First Incompleteness Theorems:
Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F
To put it in a more commonly understandable language: in mathematics, there are certain statement, can neither be proved or disproved, if the logic system we use is consistent (please refer to my previous post for what does consistent mean). It is difficult to describe the impact of this theorem. Before then, many philosophers, who often are also mathematicians, believed that the only certain knowable thing is mathematics. This discovery has turned the world of mathematics upside down, as it proved that certain mathematical statement is just unprovable.
Gödel’s Second Incompleteness Theorems:
A consistent formal system F, the statement that F is consistent, is not provable.
To me, mathematical theorems have a strange sense magical realism as Gabriel Márquez delivered in his masterpiece of a hundred years of solitude. Mathematical theorems are often stated in a plain tone, simple and seemingly unremarkable. But so many of them has transformed our understanding of the universe. From Newton’s formulation of calculus (or Leibniz, depends on your perspective), to Schrödinger picture for quantum mechanics, each discovery accompanied great shifts in our understanding of the universe. In this case, Gödel had left us with a true dilemma:
As much as we hope that to be consistent, we may never be able to prove its consistency, and we shall hope is that we can never prove that it is consistent.
Why does it even matter to be consistent? To answer this question, I will take the following example from Undefined Behavior:
Considering the following two statement –
A. No pig can fly.
B. Some pigs can fly.
Those two statements are complimentary and mutually exclusive, which means that in a consistent formal system, one and only one of them must be true. But for a inconsistent formal system, they can be both true. Using Boolean notation:
B = NOT A
Now consider a third statement –
C. Unicorn exists.
We know that A is True, so C or A is True. Because B (NOT A) is True, combining with the fact C or A is True, C must be True.
This strange logic algebra has lead us to conclude the existence of unicorn using the facts about pigs’ ability to fly. This example is constructed to be absurd to demonstrate how easy it is for us make conclusions when we are inconsistent.
In a world where Orwell’s pig is right, that “all animals are equal, but some are more equal than the others”, everything is true. But also, nothing is true.
So, let us hope, for our own sake, that we are consistent in our values and beliefs even though we might never find it out.
The end of man is knowledge, but there is one thing he can’t know. He can’t know whether knowledge will save him or kill him.
― Robert Penn Warren, All the King’s Men
RhaPsoDySea 