The barber paradox: If the barber shaves everyone who does not shave himself, who shaves the barber?
The barber paradox may strike as a trivial witty remark to many, but it is related to something far more serious. It is an example of Gödel’s incompleteness theorem, which says that formal logic (arithmetically expressible logic) cannot be both self-consistent and self-provable. In other words, the consistency of formal logic cannot be proved in the formal logic itself.
Gödel’s incompleteness theorems are often misinterpreted and misapplied to say that there is no truth in logic. But what it reveals is that formal logic is not everything. There is also intuitive logic (which is different from intuitionistic logic) that the human mind perceives.
That the human mind is capable of not only formal logic but also intuitive logic is a fact, as evidenced in the existence of statements that formal logic cannot prove, but the human mind knows it to be true by intuitive logic.
Berry paradox
The Berry paradox is an example. It can be illustrated in the following statement as an example:
“There exists a natural number n such that n is the smallest natural number that cannot be defined by a finite number of words in any language.”
The human mind knows it to be true because, due to the infinite number of natural numbers, there must be some natural numbers that are too complex or arbitrary to be defined by a finite number of words in any language.
However, the formal logic cannot prove it because any proof from the formal logic itself automatically leads to a contradiction. If such a natural number n exists, then it can be defined by the phrase “the smallest natural number that cannot be defined by a finite number of words in any language”, which automatically contradicts the assumption that n cannot be defined. This is known as the Berry paradox.
Berry paradox is also related to Gödel’s incompleteness theorem because it shows that in formal logic systems, there are some statements that are unprovable or undecidable, or more precisely, not provable without automatically causing self-contradiction.
But what is most amazing is that the human mind knows the statements are true through intuitive logic.
Intuitive logic and formal logic
The intuitive logic itself is often misunderstood to be some kind of subjective guessing work. It’s not that. The human mind can perceive or intuit abstract concepts in mathematics and logic, such as numbers, sets, and proofs. Gödel believed that these concepts have an objective existence independent of our minds and that we can access them through a kind of rational intuition.
This view, of course, contradicts that of materialists.
If Gödel is right, which he is, then his incompleteness theorem shows that mankind has been given a gift, which is a unique perception ability, namely that of a godlike independent observer of the universe.
Man is not God but is made according to the image of God (Genesis 1:26-27). No other creation has this unique gift (with perhaps angels as the only exception).
It also means that the human mind is more than arithmetic. Although it has a very strong arithmetic component, it is not all arithmetic.
The human mind is capable of both arithmetic logic and intuitive logic. The former can be programmed, simulated, and vastly enlarged with a computer (this includes AI), but the latter is fundamentally not programmable because machines, in their nature, do not have intuitive logic.
Intuitive logic and AI
AI is both underestimated and overestimated. It is underestimated for its synthetic power and functionalities of automation but overestimated for its ability to surpass and replace humanity. See, AI is not generative but synthetic.
This is because AI does not have intuitive logic. It is an inherent and fundamental limitation in its nature, not a temporal limitation of technological advancement.
Intuitive logic is not an emergent phenomenon that derives from arithmetic logic or secretes from the material of the brain. It is independent and has its own objective existence and origin. It is closely related to consciousness. See Why AI Will Never Replace True Humanity and Left-brain thinking is destroying civilization.
Without intuitive logic, AI is fundamentally different from the human mind. It can be a billion times more powerful than the human mind in formal arithmetic algorithms, but it does not mean it will become human-like.
AGI is a lie designed to deceive.
The source of confusion
Concerning Gödel’s incompleteness theorems, mathematics gives the most complex and arguably strictest explanation, philosophy gives a simpler explanation, but theology gives the simplest yet most meaningful explanation.
Confusion arises from the inconsistency of formal logic. Such confusion always has to do with the observer’s position, whether it’s internal or external. The observer’s position can be either (internal or external), but cannot be both without causing contradiction.
The barber paradox is an example. Seriously, who shaves the barber who shaves everyone who does not shave himself?
The rule creates an apparent contradiction because if the barber doesn’t shave himself, he must shave himself.
However, the paradox arises because of the dubiousness of the observer’s position. This has to do with set theory but can be simplified as follows:
Assume a set of people which constitute a set of objects of the thought experiment. Divide the set into two subsets. Those in the first subset shave themselves, while those in the second subset do not.
Now, there is Barber, who is outside of the whole set. Barber does not belong to any of the two subsets. He is independent. He acts upon the rule but is not subject to the rule. He shaves everyone in the second subset according to the rule. But he has sovereignty about whether he shaves himself because he is not subject to the rule. He is the external independent observer, a subject. He is not a member of the set of objects. There is no logical inconsistency in these conditions.
But a contradiction automatically arises when you change Barber (a specific independent subject) into ‘a barber’ (an abstract dependent object) and put him into the set.
This is because, once the barber is included in the set, you have a closed system. And as a closed system, it no longer in itself has an observer. That such a system cannot be both complete and consistent is not a surprise, much less an excuse for rejecting the existence of truth. It is a proven fact according to Gödel’s incompleteness theorems.
The Berry paradox shows the same observer’s positional contradiction. From the godlike independent observer’s position, the human mind looks from the outside of the natural numbers set and the language set and knows the Berry statement to be true. This is not a guess but direct knowledge. But once you place yourself into the human language set, you subject the statement (which carries the semantics, a product of the human mind) to a self-referencing rule to automatically cause contradiction.
What confuses people is that we, as human observers, unconsciously and simultaneously place ourselves in two different contradictory positions when we analyze these logic schemes: we are both internal and external. Being internal means being within the set, abstract and dependent, merely an object with no intrinsic logic, and cannot be an observer. But being external is being outside the set, specific and independent, with God-given intuitive logic and formal logic, and can be an observer.
Holding the above two different positions simultaneously is self-contradictory, but that’s all right because we all do this. Just don’t jump to the conclusion that truth does not exist.
Truth exists
Even within formal logic alone, there is much truth. Otherwise, mathematics and science would not exist. It just does not have the complete truth.
But people fall victim to different fallacies at opposite ends of the bar:
(1) some use the fact that there is no complete truth in formal logic as an excuse to evade truth, refuse to reason, etc.;
(2) others take math and science as absolute and complete truth.
Thevabove fallacies are either illogical or dishonest. But they have different flavors. The first takes the lack of complete truth to mean complete lack of truth, while the latter takes incomplete truth as complete truth.
Worth noting, however, is that the above latter is a form of idol worshiping and is thus particularly harmful, increasingly so in the age of AI.
See another perhaps more interesting example: The sleeping beauty paradox is a fallacy.