Gödel’s Incompleteness theorem

Elias asked:

A question similar to this one has already been asked by someone else, however it was in my opinion not answered in the original spirit of the question.

Gödel’s incompleteness theorems as far as I understood them showed that no system of human logic can prove its own consistency.

This is also obvious to common sense because every logic accessible to mankind requires a reason for everything it postulates. Therefore it also needs a reason for its own laws to be true, which cannot be given based on those laws, since those laws have to be established, i.e. reasoned to be true and existing, first.

So it seems to me that when one tries to explain reality by human logic one must conclude that there is at least one ‘Something’ (in the broadest sense of the word) which is illogical in the sense that it is not bound by the laws of human logic and therefore does not require a cause or a reason for it to be. Since this ‘Something’ is illogical and humans have only logic and empirical observation to describe reality (or guess on reality) no description of this ‘Something’ is possible for us (unless we observe it empirically).

This would show that human logic can never explain reality, i.e. answer the question ‘Why is there Something?’ and that either (A)we conclude that there is ‘Something’ transcending logic, which would not be far away from the concept of god or (B) that human logic is initially flawed (since it is not consistent) and we therefore can not know anything. Is this reasoning correct?

Answer by Shaun Williamson

Elias you have the wrong idea about all of this. Logic is not meant to explain explain the world, it simply acts as an explanation of our concept of a logically valid argument.

The theorem you are talking about is called ‘Gödel’s Incompleteness Theorem’, it is not called ‘Gödel’s Inconsistency Theorem’. The Incompleteness Theorem does NOT apply to our system of logic. The Incompleteness theorem applies to our system of mathematics. Our systems of logic can easily be proved to be both consistent and complete.

The problem in mathematics arises because our language allows us to make self referential statements. So for example the sentence ‘This sentence contains five words’ is both a true mathematical statement and a sentence that talks about itself. Gödel was able to use this idea to show that no axiomatic system of mathematics could be proved to be complete and we need an axiomatic system in order to prove the consistency of our system of mathematics. Logic doesn’t suffer from the same problem.

An alternative proof of the incompleteness theorem is contained in Alan Turing’s Theory of Computable Numbers. The American logician Alonzo Church also found an alternative proof of the same thing.

Hope this isn’t too confusing the main points to keep in mind are 1. Logic isn’t meant to explain reality. 2. Our systems of logic are provably consistent and complete 3. We can never prove that our system of mathematics is both consistent and complete unless we restrict it in some way e.g by not allowing self referential statements in mathematics. 4. We can know lots of things given the ordinary meaning of the word ‘know’. However we can also ask questions that we don’t know how to answer and that may not even make sense. It is humans who ask questions and if you ask a question you must be prepared to explain in detail why the question makes sense and what sort of answer you would accept. I am not sure that the question ‘Why is there something?’ makes any sense.


Answer by Peter Jones

I find your reasoning mostly correct. By a similar process of reasoning Kant and Hegel are led to the idea that both the extended universe and human consciousness require an original phenomenon that is not an instance of a category. The categories of thought cannot reach all the way down for the reasons that you give, and so cannot be fundamental. Hegel calls your logically necessary ultimate phenomenon a ‘spiritual unity’. Here the term ‘unity’ would indicate that in no case is it ‘this’ or ‘that’, and it is therefore beyond the reach of logic and the intellect.

However, it would not be ‘illogical’. It would transcend the ordinary world of duality where everything is always ‘this’ or ‘that’, but it would be sound logic that leads us to this conclusion. The importance placed on an understanding of dialectic logic in the Buddhist universities can be explained by the ability of logic not only to refute false views but also to betray its own incompleteness. If we cannot accept your reasoning, and thus the limits of bivalent logic, then we cannot make a systematic theory of the world fundamental. We would always have to leave something out for exactly the reasons you give. This is Russell’s paradox, the reason why he could not axiomatise set-theory. The attempt to make any bivalent logic fundamental leads to intractable problems of self-reference.

The question of whether we can acquire certain knowledge by the use of logic is easy. Logic produces only relative truths and falsities. This need not be a pessimistic view of knowledge, however, for it is possible to know things without any dependence on logic. You might know you are in pain, for instance. For the third time today I’ll quote Aristotle’s crucial observation that ‘true knowledge is identical with its object.’ No mention of logic.

As you say, this analysis of the limits of logic leads us to the idea of something that seems rather like God. But it can be seen that where philosophers undertake this analysis they usually call this phenomenon by a different name such as Tao, Nirvana, Ultimate Reality, Unity, Godhead, Bliss, the Authentic, the Undifferentiated, the First or Original. Plotinus uses the term ‘Simplex’, which indicates its lack of conceptual complexity, the idea that it lies beyond the categories of thought, or, in the words of one Christian mystic, ‘beyond the coincidence of contradictories’.

But this would not mean we cannot know it. We can know it intimately if we are it, and we must be if, as logic suggests, Reality is unified at an ultimate level. For this view we need not abandon logic, but we would need a logic of contradictory complementarity. Hegel’s idea of ‘sublation’ is important here. If you are mathematically-minded then for an example of how such a logic would work you might like to read George Spencer Brown, who solved Russell’s Paradox in his book ‘Laws of Form’.

Your final question asks whether human logic is flawed because it is not consistent. I would say not. It can be consistent just as long as we do not imagine it is complete. It is only when we imagine (in either ontology or epistemology) that the duality required by the functioning of our intellect reaches all the way down that our logic becomes inconsistent and flawed. This was Russell’s problem. He did not know (or want to know) religion well, so did not have the principle of ‘nonduality’ at his disposal and could not transcend dualism for his philosophy or mathematics. If, however, we accept that the universe is not the set of all sets, which would be a paradoxical idea, but something Kantian that is entirely beyond sets, then our system of logic can be consistent from the ground up, as is demonstrated by Brown with his ‘calculus of indications’.

If you wish to pursue these issues then a directly relevant experimental essay is ‘Exploring Connections: Music, Cosmology and Mathematics’ at http://theworldknot.wordpress.com/


One thought on “Gödel’s Incompleteness theorem

  1. Elias: Gödel’s incompleteness theorems as far as I understood them showed that no system of human logic can prove its own consistency.

    Shaun Williamson: The Incompleteness Theorem does NOT apply to our system of logic. The Incompleteness theorem applies to our system of mathematics. Our systems of logic can easily be proved to be both consistent and complete.

    I am inclined to add, that Godel’s theorems are theorems of ‘mathematical logic’, technically. Then, here, I’m not sure that the distinction between our systems of logic, and our system of mathematics, is utterly perspicuous. But, it might help to keep in mind that mathematical logic is a subfield of mathematics.

    Meanwhile, what about the assertion that our systems of logic can easily be proved to be both consistent and complete? Is second-order logic complete? No. But, one might say here, that a class of formal logics that have been most intensively studied and most widely used, is complete. One might also say, that the question wasn’t about semantic completeness, –so I won’t even add a definition of what it is. What about consistency? Well, In classical deductive logic, a consistent theory is one that does not contain a contradiction. Still, one has to add something about how the lack of contradiction can be defined. The sense used in traditional Aristotelian logic would be, more or less, half of the picture of what we mean by ‘complete’, and is what is also referred to as
    satisfiability. One might slow down, here, though, and say merely, that for classical logics, it is *generally* possible to re-express the question of the validity of a formula to one involving satisfiability. At the same time, satisfiability is undecidable and indeed it isn’t even a semidecidable property of formulae in first-order logic

    Really, my only point in all this, is to give you a hint, that when the incompleteness theorem is discussed, it is easy to find more enthusiasm than competence, as such discussions tend to be loaded with inadequacies and errors.

    I’m not at all certain, in fact, that the demand for a knowledgeable and reliable exposition of the incompleteness result has been satisfied, ever, anywhere. I don’t aim to fill this need, as I know that I could not succeed outstandingly. Rather, I encourage you to consider trying to get through life without an adequate understanding of the content of Godel’s
    theorem, and also neither of how it is proved. That there aree especially some naturallyoccurring misunderstandings, that I do not intend to correct here, is in the end all that I can offer you.

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