This article is the second of a series of 3 about Formal Logic and Religion. Find the first one, introduction to formal logic, here.
I will now try to introduce you to what is arguably the most important result in formal logic, Gödel’s incompleteness theorems, and deduce a constructive proof of the existence of God.
Warning: This is going to be a very informal discussion, but there’s a plethora of better writing on the subject if you want to explore this deeper, which a quick Google Search should help you find. It’s one of the most discussed topics in mathematics.
What is it?
In the previous article, I gave you the basics to understand formal logic, by focusing on sets of beliefs containing a contradiction and see that they were all equivalent. Let’s now look at the other ones. A set of belief that does not contain or imply a contradiction is called consistent.
Godel proved that whatever your system of belief, there are statements that cannot be proved by it. The proof is actually not that complex, though I never understood it until I read some kick-ass vulgarization recently: Godel proved that in any system of beliefs, you can use the basic principles to express a statement similar to « This sentence is false » that cannot be proved to be either true or false.
As a follow-up to this result, Godel also proved that you can never prove that a system is consistent with the principles of the system. The proof is a bit more subtle but revolves around the fact that if you could, you could use that proof to prove that « This sentence is false » is true, and that’s absurd.
What does it mean?
Of course, Godel was talking about math stuff. The « system of beliefs » he was talking about was mathematical axioms like [1+1=2, you can always pick a random element in an infinite set…]. So you see that the beliefs I’m talking about can be very obvious and non-arbitrary. But the arguments hold whatever the system.
These theorems have huge implications for reasoning in general. It’s a formal proof that whatever you adopt as system of beliefs, there are things you cannot prove to be either true or false, and in particular you can’t prove that your system of beliefs is not inconsistent.
I think, if nothing else, this forces you to be humble vis a vis your beliefs, no matter how obvious and indisputable they are.
« There are more things in heaven and earth that are dreamt of in your philosophy. »
Transcending the system
So any thought system has necessarily shortcomings, and furthermore you can exemplify the limits of the system using the elements of the system. I like how this idea echoes the classic trope that every system contains their own undoing.
This article by SpeculativeWeeb is a really cool take on Godel’s theorem applied to Puella Magi Madoka Magica. It highlights that Madoka essentially found this shortcoming of the system, the « this sentence is false » of her own world. She forces it to realization using her wish to Kyuubey. In a nutshell:
She wishes for all witches to vanish before they’re even born. However by doing so she becomes herself a witch, so she vanishes and can’t make that wish.
She exploited the shortcoming of the system in order to break it. The only possible resolution is to ditch this system, and a new one replaces it that manages the problematic element (a world without witches and without Madoka).
However, the new system is also bound to have a transcending element, which is what Rebellion tried to tackle with more or less success. Whatever you do, you can’t escape Godel… There’s no perfect system without transcending element.
Managing the transcendence
If any system contains their own undoing, some have certainly tried to manage this necessary shortcoming to make it foolproof.
The Matrix is an interesting example: machines first tried to build a utopia where everyone was happy, but a flawless system was bound to fail. Instead, they had to include faults in their system: they added unhappiness inside the Matrix to make it stable.
But of course as a system, this also had its shortcomings and had an element that could transcend it: the One. So the machines actually managed a meta-system which included the existence of a transcendental element as part of the plan, a chosen One who would have to make a dummy choice to keep the ball rolling. But hey, this is a new system, so it has to have something that can transcend it…
It’s not uncommon in this context to see the smartest systems try to include and manage their own undoing in such a way. There is countless examples in sci-fi, like The Giver, or Westworld. « ‘the plan fucks up‘ is an element of a bigger plan » is a classic trope in fiction. Note how it builds up on meta.
But no system does it quite as well as the real world. Indeed, the genius of neo-liberalism is to plan for this element of contingency, and to include the resistance to the system as part of the system. Everything can be monetized, even anti-conformism.
You can find more information on this trail of thought all around the webs, like this brilliant video for example:
Implication for the nature of the universe
What about the implications of the second theorem to the real world? If you can’t prove a system’s consistency from within the system, does it mean that we’ll never be able to prove formally that the world is deterministic? Does it mean that we can’t prove whether or not we’re in a simulation?
Arguably, it doesn’t really matter, because the world will be the same whatever you believe. Life will still follow deterministic patterns even if you can’t prove it. But it’s an interesting echo of Hume’s experimental philosophy. He argued that just because things have always happened a certain way doesn’t mean they’ll keep happening, and there’s no reason why the world couldn’t suddenly stop. If we are in a simulation, maybe the computer will stop, or change the parameters… How would we ever see that coming? Maybe this ambiguous report of causation and correlation is the transcendent part of our reality.
Everything could suddenly crash. But it won’t. That’s just how the world is. But maybe you can’t ever prove it. That’s intriguing.
Proof of God
Interestingly enough, as it pertains to our reflection about logic and religion, Godel was very proud to have proven the existence of God mathematically. Unfortunately, it is an ontological proof and is therefore total garbage.
However, Godel did prove that whatever the system, there is inherently something that transcends it. And that this something is contained within the system. I’m willing to let this be called God, for all the chaos and confusion that it will surely bring, even if it’s just a glorified alias for the logical concept of « This sentence is false ». In fact, let’s call that God-L, because it’s fun.
We’ve proved that whatever the system, it’s by nature incomplete. This incompleteness is God-L. There is always God-L, it is absolute. Furthermore, it’s true for any thought system, so it’s also true for a system that tries to encompass this fact. If you add God-L to your system, there’s still a God-L that transcends it (as we saw in the Matrix). What we want to call God-L is in fact the union of all these God-Ls, the infinitely meta-transcendence of all systems. But it is still incomplete and transcendable… Which makes it the perfect transcendental element of a meta-meta system that tries to reason about systems, which brings me back to my fixed point of meta…
God-L is the very essence of incompleteness and unexplainability in the universe. Instead of being an all powerful wishgranter, it’s by nature lacking. Maybe it’s a nice tool for your spiritual health…