Nobody seems to know a lot about a person named Kurt Gödel – a 20th century mathematician and philosopher – but in certain circles, he’s regarded just as high as Einstein for his invaluable contributions to logic in general, particularly modal logic of second degree, and most of all for his work on the foundations of mathematics – modern set theory. That does not have to tell you anything in particular – and this post is not about his many extremely complex and highly sophisiticated theorems. Just for refernence and example, he managed to prove god’s existence, at least according to a number of highly renowned people in the field. They had to employ computers to make up for missing links in the fragment of the proof that Gödel left us, but they believe they have plugged the gaps sufficiently to be able to say, with a high degree of confindence: yes, Gödel managed to prove God’s existence via pure logic.
Yet, the man is most renowned for his incompletness theorem, published in 1931. This article aims not to give you the exact mathematical details of the proof, but rather tries to communicate it as concisley and in its purest essende as humanly possible.
I assume you know of the liar paradox, most famously put forward by Bertrand Russell in the form of a short sendence: “A man says: I am lying.” This sentence creates an infinite cycle of deductions, each contradicting the other. If it is true, it is obviously false, as it claims to be a lie. If it is false, then it is in accordance of what it claims: not to be true, which in turn makes it true.
There is no generally accepted solution to the liar paradox. Many have tried to solve it, all have failed. The usual way of trying to deal with it is to claim that somehow the sentence is malformed – in some way. For example, it has been claimed that no self-referential sentences are to be taken seriously, and must be excluded from valid use of language altogether. The problem with this approach is, that it also excludes many innocuous sentences like: “This sentence has five words.”
In essence, all the attempts at solving the liar paradox have failed – which is why it is sometimes considered a “true” paradox, an antinomy, a sentence that is paradoxical but cannot be solved. It is transconsistent (or paraconsistent, which means the same thing), i.e. it transcedes the consistent, that is contradiction-free realm of langauge.
Why am I saying all of this? Because Gödel found a very clever way to employ the liar paradox – and it sent shockwaves through the mathematic community when it was published, as it claims that mathematicians have to decide: they can either accept that their field is riddled by an inconsistency – something which mathematicians can never accept, because being free of any contradictions is the hallmark at the very core of maths – or have to acknowledge that there are certain truths in maths that never can be proven to be true. How did Gödel manage to back all of mathematics into such a corner?
Consider the sentence: “This sentence is true and cannot be proven.” The standard way of determining if a sentence is true or not is to look at the implications from both sides: what follows if it is true, what follows if it is false. If this sentence is true, then it states that there are true statements that cannot, by design, ever receive proof of their truth. If it is false – well then it could be proven, but then again, a falsehood can be proven, which would in turn bring down the entire system, if its proving mechanisms lead to a proof for something that is false.
The whole story is more complex than what I am conveying here. Gödel had to invent an elaborate numbering scheme, and establish a so called lemma (a lemma is something like a stepping stone, a prelude to the proof of the theorem), in order to make his proof watertight. I am not going to bore you wth those details – just please accept, that it is generally accepted, that he succeded in doing so, and formulated his proof in the terms of axiomatic set theory, which in turn is the commonly accepted basis for all maths on the planet, all maths as we know it.
So a seemingly innocuous sentence brings down all of mathematics? Not quite. Not long after Gödels proof, a whole book was written as a rebuttle of Gödels incompleteness theorem, which in turn divided mathematicians, who are willing to to look at the foundation of their science, into two spheres: the ones that believe Gödel was right, and the other block that insists he was wrong somehow, but can’t quite explain away his theorem.
I wrote a book in which I tried to transfer Gödels incompleteness into philosophy, and generalize it to a degree that it applies to many philosophic theories. Its relevance hence lives on, and so does Gödels legacy. When I get around to it, I will write a synopsis of that transfer and link it here for the interested mind to study (and critically question it).
For now, I’d like to leave you with the thought that even mathematicians are bothered with riddles and inconsistencies – just like any human being has to endure, for example, when feeling contradicting emotions. The thought can convey a calming feeling: no human enterprise is free of contradiction in terms. All we can do is try to adhere not to being free of contradictions, but try to be as consistent with our selves as much as possible, rather than display consistent traits, repeating patterns, or even self-enforced rules – because all of them may have at least one exception. If maths, the pinnacle of consistency, cannot hold its own, then we do not have to even try, or maybe even should not try to achieve consistency in our thoughts, deeds, and beliefs. I personally don’t think that is what life is about – but I see a lot of people who seem to willingly but that burden onto their backs, trying to show off a consistent, never changing face on the outside.
This is far from a well-established consequence of Gödels theorem – I am just trying to convey a personal thought, an opinioin. Maybe consistency is not something worth to be sought for with so much effort: maybe it is something that is just nice to have, but not a prerequisite, an axiom to anything, at all.
What do you think?