Godel is curious about absolute truth. And this makes sense: he’s an avowed Platonist, so he believed that numbers possess a sort of hyper-reality. Most of us, in fact, also subscribe to some form of this belief–the precise formulation varies, but more or less it amounts to the observation that the proposition that “two plus two equals four” can be said to make sense regardless of the underlying material reality. As long as the symbols are taken as such, it makes sense: the proposition actually only refers to the symbolic coordinates of representation. This is the deep point Godel wants us to get: the symbolic coordinates we use to ‘construct’ an image of our reality are always, inherently incomplete.
Moreover, he’s able to prove they’re incomplete, and for a rather ‘banal’ reason. But first it’s important to understand how he demonstrated this. It’s good to actually look at his proof, though without significant mathematical training it can be quite difficult. However, there are several pages of introduction and conclusion which are quite lucid, and the symbols are actually relatively few. Furthermore, he constructs his entire system from scratch–‘out of whole cloth,’ as it were. But how does it work?
First, he builds a system for performing basic arithmetic operations and assigning numerical equivalences. Further he constructs a numbering system so that every ‘proposition’ of the system can be given a unique number (called its 'Godel number.) Then he asks us to imagine a function which, given such a Godel number, would tell us whether the proposition can be proved within the system. Finally, the ‘completeness’ question amounts to: given a formal axiomatic system, can all the true propositions in the system be derived from the first principles of the system?
The answer is no, but it’s important to understand why this is not possible. Let us suppose that we have got such a perfect, completed system. In this case, it is true that the system is complete. Therefore, this statement itself has a Godel number, as it is part of the system itself. But this statement cannot be proven true by any other statements in the system. Now we’ve got a ‘rogue’ true statement–that is, which we believe we know–which still cannot be proved by the system. There will always be these nomad truths which immediately escape formalization. For example, we can simply create a bigger system which accounts for some (or even all) of the nomad truths unclaimed by the system. But then the proposition that this ‘new’ system is complete is still unaccounted for!
So why are they incomplete? Because there’s always additional situations we can percieve or recognize to be true which are not yet covered by our symbol systems. Thus, one way of phrasing Godel’s discovery is that it is impossible to formally establish the internal consistency of a very large number of commonly-used deductive systems. He demonstrated that once a system becomes complex enough to allow for self-reference, completeness is no longer a meaningful category.
What does this mean, then, about absolute truth? Well, a few things. We can combine this with Heisenberg and Einstein and say that ‘truth’ is characterized by three features: uncertainty, relativity, and finally–incompleteness. In fact, this is what it means to be engaged as a subject–that is, we only find the truth when we struggle, so we can only experience truth as a partial assemblage of identity, a marginal structure… Godel thus shows us there can be no truth-machine: or conversely, that truth is an uncertain mechanism.