Gödel's Proof of God's Existence
arxiv.org
arxiv.org
– The basic modal logic K is sufficient for proving T1, C and T2.
– Modal logic S5 is not needed for proving T3; the logic KB is sufficient.
– Without the first conjunct φ(x) in D2 the set of axioms and definitions
would be inconsistent.
– For proving theorem T1, only the left to right direction of axiom A1 is needed.
However, the backward direction of A1 is required for proving T2.
As someone who's read the first 2/3 of GEB at least twice, I'm planning to reorganise my entire worldview around this.
Edit: Seems like this is the code - https://github.com/FormalTheology/GoedelGod
Great books, unfortunately never translated from Swedish. The story is also about whether information can be destroyed, which of course was a hot question around the time of Nilson writing the books (they were released in the first half of the 90s, I think).
A little deviation, but I just wanted to underline how strongly I feel, again like you, to bring this into my worldview! I need to study the scriptures! :-D
Start with a selective, loaded definition of "god" that seems to amount to "A god is a being who must exist."
God is all that is good; not existing is not good, therefore god must exist.
QED.
Well, you can't really win here. It's either a hand-wavey definition you could never formally prove, or a rigid, 'loaded' definition.
That said, the basis of the proof does seem to be that 'if god exists, then he must exist', which is not really groundbreaking.
Actually, its "if god must exist, he exists".
One thing to note from the axioms is that for every property either that property or the negation of that property (but not both) has the label of 'positive'. This is already pretty strong, because it's making claims the about the 'positivity' of properties that are completely unrelated to the rest of the argument. It's almost just a labeling game: given p and -p, you must label exactly one as 'positive'. Is the property of being red positive? Saying no is equivalent (according to these axioms) to saying that the property of not being red is positive.
You can pick one model for the axioms where being red is 'positive', and then the God that's shown to exist will be red. You can pick another where being not red is 'positive', and then the proof implies that God will not be red. It's not a logical contradiction because they're different models, but it certainly makes one wonder what the proof is talking about when it mentions 'positivity'. I have a suspicion that the axioms themselves generate an inconsistency and hence they could generate proofs for any statement.
Anyways, this paper is about formal theorem provers, not theology, so hats off to the researchers implementing higher-order logic.
A5 Necessary existence is a positive property
Oh, yeah, that's not circular at all...
Such is the world of philosophy. Remember that some of the greatest scientists and mathematicians lived when all of it was just philosophy. Philosophy is a training exercise to think imaginatively and some of our best science stems from an imaginative spark. It's like saying that people professional athletes running around a field (philosophy) is pointless because they are not running in a race for gold (science) at that specific point in time.
This is by all means not scientific, precisely because you can't address that specific subject matter scientifically:
1. God exists if he makes himself/herself visibly and measurably known and is observed and measured by a human being and the observations are confirmed by peers within the scientific community. 2. God does not exist if a human is present to see the end of the universe (or variant there-of) and he has not been yet observed (assuming he/she is tied to the existence of the universe) and the lack of observation is confirmed by peers within the scientific community.
We simply don't know until either (1) or (2) is satisfied, therefore this is exactly the type of stuff philosophy is good at. It's not saying "can we prove it" it's saying "let's think about creative ways to prove it if we can get a little lax about the scientific method." Were "getting lax about the scientific method" means that "proof" can be derived from either common sense or logical arguments.
Not really. Galileo and Newton turned philosophy into science by doing experiments, and scientists never looked back. Science differs from philosophy by being steered -- defined -- by empirical evidence, and by falsifiability, the idea that any idea must be testable against reality and must be discarded if the test fails.
That's certainly not philosophy.
> It's not saying "can we prove it" it's saying "let's think about creative ways to prove it if we can get a little lax about the scientific method."
There's no middle ground such as you're suggesting here. There's no overlap between science and philosophy. They are distinct ways of processing ideas. One of them must be compared to reality at every turn.
Important to note Gödel's completeness theorem here!
You might be even more interested in Prolog, a language which has a close relationship with this stuff: "Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is declarative: the program logic is expressed in terms of relations, represented as facts and rules. A computation is initiated by running a query over these relations." (http://en.wikipedia.org/wiki/Prolog)
This is why everyone should take an AI class.
ITYM incompleteness theorems.
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_the...
There are obvious (and not so obvious) problems with such an argument even when formalized, some of which are discussed on the Wikipedia page.
Logically proving the existence of God is a very old game. Modern atheists tend to ignore these proofs in favor of attacking superstition and the more obvious theological nonsense.
Which is a shame, because they can be refuted, with interesting consequences for one's worldview. The stance that most powerfully refutes any Ultimate argument for God (of which there are a few good ones) is, imho, Madhyamaka Buddhism, which holds that everything is empty because all phenomena dependently co-arise.
Made me think of a Uni course on Thomas Aquinas from eons ago. I'll have to go back and check but I think:
"An essence of an individual is a property possessed by it and necessarily implying any of its properties"
could have come from Aquinas' "Being and Essence".
And - forgive me it has been eons - didn't Wittgenstein deal with this time of formulation showing you could in parallel construct its opposite.
I've only just begun studying Gödel's proof of incompleteness and so I'm not qualified in any way to verify this paper. However it seems to me, as a layman, to be a very large stretch and mostly a tongue-in-cheek exercise of various logic proof verification software.
Or possibly a troll. Hard to tell. :)
I'm not an atheist, but for truth's sake you would notice that Gödel definition of God is very wide and not necessarily aligned with any mainstream religion.
> Axiom 5: Necessary existence is a positive property
That's in total disagreement with evolution. No being is really necessary.
> Corollary 1: The property of being God-like is consistent.
Now that's in huge disagreement with any religious book, as all of them are full of inconsistencies.
Any proof lacks any value if the starting axioms lack value by themselves.
Newton wasted years of his life studying the bible. We shall simply view all this as the waste of time it was and simply expect future genius to not do the same.
It's like discussing the concept of an honest politician -- such a thing can obviously be proven to exist in principle, but that doesn't require that it exist in fact, in everyday reality. A good thing, too.
All other issues aside, the Holocaust proves that either (a) God doesn't exist, or (b) he's Spinoza's God, the one who can't be bothered about humans and their problems.