# How can the logical impossibility of infinite self reference be reconciled in the case of God?

This assumes that God conforms to the laws of logic as without doing so we can make absolutely no deductions about him, his existence, or any of his actions or attributes.

Within computer science, there is the notion of computability. This is the idea of whether it is possible to calculate something or not. It is provable that there exists a set of problems that cannot be computed. See the Halting Problem

If we take the attributes of the traditional Christian god, that he is all knowing, and all powerful, it appears this creates a logical contradiction. Here is an example:

1. Event A causes event B within the universe.
2. God decides that he does not want event B to occur so he stops event A from happening.
3. Event A no longer happened and therefore God would never intervene in the first place. Go to 1.

This is a very simplified example, and suffers from the problem that it is grounded in the concept of time, and this would imply God is able to circumvent this paradox because of the fact he is outside of time.

However effectively the same occurs with the halting problem, except time is not an issue. If God evaluates the universe (all of space and time) and comes to a decision, he takes action but some actions will depend on what he originally observed meaning it must be re-observed and re-evaluated. In this 'new round' of observation he is now evaluating himself and his previous actions. In some cases, this can lead to infinite self reference which is logically undecidable.

This seems to leave us with 3 options:

• God is not all powerful. He could be very close, but with the diagonalisation proof I think we can show there are conceivably situations that would not work.
• God is not all knowing. Again, he could be very close to all knowing, but can't logically be entirely all knowing.
• God does not conform to the laws of logic.

I would be really interested to hear any thoughts you may have on different parts of this. If anyone here has studied the halting problem in more detail than I it would be interesting to see where it doesn't hold.

Edit: I feel I should make a small clarification about the logic assumption. If you actually know what logic is, and I don't mean "things feeling scientific" or something wishy washy like that, I mean the laws of logic, then I think you will realise that saying God does not conform to them effectively makes the concept of God pointless.

I know Christians who agree and disagree with this, but it seems that the pattern is those who actually understand what logic is agree that God must conform to it.

It is possible to say that God does not conform to logic, however if you believe this you must also accept that, from our point of view, we can then say nothing about his nature, his actions or his existence.

I don't think this path of reasoning benefits anyone in any way.

Edit 2: Here is a short description of the halting problem. The analogy I am making is that God is essentially one of the 'oracles' referred to in the article, and after acting within the universe must re-evaluate the universe to see if more changes need to be made now that he can see the results of his actions, thus leading to self-reference.

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God doesn't conform to the laws of human logic... :P (Isaiah 55:9) – El'endia Starman Apr 9 '12 at 2:57
@El'endiaStarman There are very few base laws on which our entire logical foundation is built. If God does not conform to these, he both exists and doesn't exist at the same time. If he doesn't conform to the base principles of logic that humans are bound to, we can know precisely nothing about him, and the entire discussion has no point. – danpalmer Apr 9 '12 at 3:04
your whole argument (which, by the way, I find ironic since it depends on logic - can't be helped, of course) relies on the premise that God is an observer. You're trying to explain God in human terms. The psalmist tried that and had to give God hands and wings to describe him - physical attributes God obviously didn't have. You can't accurately describe God and his attributes in a humanly logical form. God's ways are higher than our ways. `</rant>` – Thomas Shields Apr 9 '12 at 4:14
Why is this being down-voted? I see some logic errors but it is worded well, and the question is very clear. – San Jacinto Apr 9 '12 at 12:43
I agree with @SanJacinto that this question doesn't merit down-voting. I do agree, as others have said, that the logical contradiction you're using right now isn't very effective for representing the thing you're really driving at, but I would love to see the inquiry get refined so that the question can be better addressed. – asfallows Apr 9 '12 at 13:00

Nothing to do with religion / God / whatever, but there seems to be a fundamental flaw in your argument here.

• Event A causes event B within the universe.
• God decides that he does not want event B to occur so he stops event A from happening.
• Event A no longer happened and therefore God would never intervene in the first place. Go to 1.

The third bullet does not follow unless you are thinking of God as Marty McFly hopping backwards and forwards.

If I have misinterpreted the logic in the question, please let me know. However, let's put this into more common terms, following your reasoning:

• smoking (A) causes cancer (B) and costs money
• I decide that I don't want to get cancer or waste money, so I stop this from happening to me by not smoking
• I didn't smoke (!A) and I find I don't have cancer (!B) (based on a statistical average) and have spare change in my pocket, therefore my efforts were wasted and there was no need not to smoke

or:

• leaving the hob gas on unlit (A) causes houses to explode (B)
• I decide I don't want my house to explode, so I check the gas is off when I'm not using it
• the gas wasn't on (!A), and my house didn't explode (!B), therefore my caution was unneeded and there was no need to turn off the gas

Even in Human logic, the third bullet is nonsense. It really isn't my intent to defend the religious view here, but this is not a sensible argument against God, and is trivially dismissed even when considering a person with natural knowledge. If we pre-suppose all-knowing, then that not includes possible consequence, but rather: absolute knowledge of consequence (and prior reasoning).

It seems that even in the knowledge that A (and B) hasn't (haven't) happened, having A continue to not happen (to avoid B) is still necessary.

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As I said, the exact example I gave isn't the best because it relies on time. If you know about the halting problem, you will know that it is not based on time, however this was the simplest version I could give in a question. – danpalmer Apr 9 '12 at 15:58
@Danpalmer: Marc's one of the highest rated users on StackOverflow. I think it's safe to assume that he knows about the Halting Problem. – Mason Wheeler Apr 9 '12 at 17:37
@danpalmer was that comment perhaps intended as for Joshua's answer? I discuss neither the halting problem nor time (although I am familiar with both). Well, I did very briefly, like for about 10 seconds between v1 and v2 and then removed it. – Marc Gravell Apr 9 '12 at 18:19
@danpalmer re "laws of logic" (in your question edit), I took degree level math - I'm very familiar with formal logic – Marc Gravell Apr 9 '12 at 18:24
@danpalmer perhaps the real unanswerable question is: does omnipotence trump halting? I'm not "invested" in a view (since I'm atheist ;p), but IMO it kinda has to - it already (if you take the Christian definition) trumps quantum theory and things like Heisenberg uncertainty theory. – Marc Gravell Apr 9 '12 at 20:19

There are an awful lot of assumptions here.

1. There is but one way to stop B
2. There is no way to stop B if A has happened
3. Etc

But it mainly boils down to this main flaw in the logic:

Event A no longer happened and therefore God would never intervene in the first place. Go to 1.

Being all knowing God would of course be able to know the possibility of A causing B. Thus the possibility of B happening is cause enough to stop A.

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This does not depend on these assumptions when you consider more complicated examples. I wish I could give you some, but to really explain this concept does take a good hour or so of discussion. – danpalmer Apr 9 '12 at 16:00

I believe you miss a few key points about the halting problem.

1. Just because there is no algorithm that can decide for every input program if it will halt, does not mean that no algorithms exist for any given input. See: Are Impossibility Proofs Possible?- Halting Problem for examples.

You assume that God would have to function over every single conceivable input, where it may be that the number of inputs itself is fixed to some subset.

2. The problem can be reliably solved by people for a given subset of programs. The wikipedia introduction to the Halting problem gives two trivial examples of such.

The halting problem relies on the fact that no algorithm could examine every branch of any significantly complex program, similar in a sense to the inability of the computer to compute every legal move in a chess game.

Simply assuming that God is all knowing immediately demonstrates that He would know every branch of execution for the Universe and would by definition be able to decided if it would halt or not.

3. Your assumption relies on the fact that once God acts He would need to evaluate the outcome.

This has never been a requirement of the halting problem that I am aware of. Nothing requires the output of the algorithm to be fed back into the algorithm, but even if you can demonstrate a reasonable formulation of the problem that meets that requirement, God is not required to act. Meaning that Ge is not required to intervene to make the Universe more to his liking.

4. There exist modifications to even simple Turing machines which, as it turns out, enables them to solve the Halting problem.

As the following Stanford article on Computation in Physical Systems states

This allows infinitely accelerating Turing machines to compute functions, such as the halting function, that are Turing-uncomputable.

So, simply by removing time as a factor, computability limitations are removed. Clearly refuting your claim that "if you know about the halting problem, you will know that it is not based on time".

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In response to the first point: you only need one conceivable input, which we can show must exist, for the idea to hold. – danpalmer Apr 9 '12 at 20:03
2. This is correct, most problems can be analysed, and leads me back to the previous comment. – danpalmer Apr 9 '12 at 20:05
3. He must evaluate the outcome in order to determine what actions he would take on the new state of the universe. I think this is exactly the problem of the 'oracles' in the article I linked to. – danpalmer Apr 9 '12 at 20:06
4. Now this is a really interesting point, and the sort of answer I was hoping to get. I haven't seen this before and so I will read up on it and get back to you! – danpalmer Apr 9 '12 at 20:07
@danpalmer I would argue instead that one must prove that the universe is not say written "use[ing] only for loops, for example, with fixed bounds"... at least from God's point of view. If on were going to apply the Halting problem to disprove His existence. – user359 Apr 9 '12 at 20:08

You're assuming that self-reference is required somehow, that God has to "evaluate himself" (whatever that means) when making decisions, and that his decision-making is based solely on observation and not on any sort of predictive ability. There's a much simpler conceptual model:

"Something important is about to happen. There are three possible outcomes. Predict which of the three is most favorable--which will necessarily be a correct prediction due to omniscience--and cause that one to occur." No self-reference required.

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God is outside of time and therefore already knows what will happen, no predictive ability is necessary, but after a change has been made, he must 'evaluate' or work out what that change causes and see if he will make another change. You could instead say that if he wants to make a change he must 'predict' or work out what will happen as a result, and that in itself may require him to consider his own actions. - As I said, this wouldn't always cause infinite self reference, but it will do in some cases. – danpalmer Apr 9 '12 at 20:14
@danpalmer but why must he work it out? He knew the outcome before he "tweaked" it, otherwise you cannot claim an omniscient God. I'm personally not convinced that He has to tweak anything, but I'm trying to play by the rules of the question. – user359 Apr 9 '12 at 20:32
@JoshuaDrake I see your point, but when I think through the process of knowing what would happen it relies on 'previous data', which creates the self reference. – danpalmer Apr 10 '12 at 3:31
@danpalmer If knowing `relies on 'previous data'` then no author could write a story, and for that matter wouldn't the Big Bang have the same issue? Everything that we know about, either hard science or the most remote religious signifier must have come from somewhere. You appear to be applying self reference in a very strange way. – user359 Apr 10 '12 at 13:23
Ok, bad wording, it is more that it relies on data about previous actions you have taken. It's like an author writing a book about himself writing a book about himself writing a book about him... etc. – danpalmer Apr 10 '12 at 20:55

The God of the Bible states "I declare the end from the beginning." In comparing God to an algorithm, you go against the fundamental claims of God --that is, you create a straw man to knock over.

God is not an algorithm, waiting for an asynchronous user event or for some unseen input data. God is the author of the data and thus by necessity of being its creator, it is sanitized to be within the exact ranges on which He wishes his state machine to operate. God has created the algorithm to run on the data. God has created the machine on which the algorithm runs and into which the input data goes. The analogy is broken from the beginning.

Edit: What you're arguing against in your question is a form of Molinism, not historical perspectives of Man's free will and God's sovereignty. In this regard, you are entirely correct which is why I struggle finding correctness in Molinism.

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Ok, fair enough. In that case we don't have free will. – danpalmer Apr 10 '12 at 3:24
@danpalmer "Free will" is such a loaded term that I can neither agree nor disagree. If you mean "liberty to effectually cause desires within myself" then I would of course agree. Not one person can do that regardless of whether or not one believes in the Biblical God. If instead you are referring to "liberty to act on one's desires" then I would disagree vehemently. What you are not considering in this case is the concurrent nature of God's acts and plan with our desires. As you can see, it's a little more complicated than just tossing out the term "free will." – San Jacinto Apr 10 '12 at 10:42
I would disagree, I think this can, with some proper consideration, actually happen entirely concurrently. I would also say that free will means the ability to do what you want. If you can't do that, or if God in any way pre-determines it, you don't have free will. – danpalmer Apr 10 '12 at 20:11
@SanJacinto For a given action, describe (perhaps in chat) how someone would be able to not act (that is, to have liberty to act or not) on the final desire that ultimately causes the action? – Chelonian Apr 10 '12 at 22:55
@danpalmer "ability to do what you want"..precisely. You don't do what you don't want. You always act according to your strongest inclination. In this regard, there is no free will (even in a non-theistic system). You cannot control what your desires will be. However, you are always free to choose that which toward you are most inclined. In this regard this is free will (even in a non-theistic system). I don't have the time to delve into it, but the theological problem is "how can God ordain my actions while not necessitating them?" and various theologians through the centuries have answered. – San Jacinto Apr 11 '12 at 12:48

It is, I suppose, conceivable that God is not bound by the laws of logic. But if true further discussion is impossible, as how are we to consider the possibilities except by using logic? I can't imagine how I could prove this either way. Will I present a logical proof that logic always applies? But if logic doesn't apply, then the proof is invalid, and the argument is circular. It is even more difficult to see how one could present a logical argument that logic DOESN'T always apply. So let's accept that God is bound by logic as an axiom.

As others have noted, your paradox example doesn't really work. You appear to be invoking the classic time travel "Run-over grandfather problem": What if you went back in time and killed your grandfather before he had any children? Then you would never be born, and so there would be no one to go back and commit the murder, so your grandfather would live, so you would be born, so there is someone to go back, etc.

But in your example, event A does not cause God to exist, or even cause him to act in a direct sense. Rather, it is the consideration of the consequences of A that causes God to act. If I shoot myself in the foot, it will hurt really bad. Therefore, I do not shoot myself in the foot. I don't need to shoot myself in the foot to know the consequences. I can learn the consequences from hearing of the experiences of others who have shot themselves in the foot, or I could deduce those consequences by considering the nature of bullets, flesh and bone, etc.

Suppose that God is outside of time. He sees all of history as one big panorama. We mortals only perceive a slice of it at any given moment, a slice that moves across the panorama as time passes. But from God's perspective, it's one big picture. He can then draw that picture, considering the implications of every line he draws, extended out as far as he wishes. A human artist drawing a picture of, say, a landscape might say, If I start drawing the base a tree here, I'll need to complete that tree upwards to the top branches, or perhaps draw it abruptly ending in a stump, etc. But he doesn't have to agonize through circular logic like, If I draw the trunk, I'll need to draw the branches, but if I make the branches too wide, then I'll have to redraw the trunk, but if I redraw the trunk ... What would be the catch? He would just draw the tree the way he wanted it. Even if we assume that God can change his mind after seeing how it turns out, so what? So he erases the tree and redraws it the way he wants. He can consider how the tree fits into the landscape and design one big picture meeting all his desired requirements. There's no need to be circular about it at all.

If we assume that God exists within time, but is all-knowing and all-powerful, it doesn't change the model much. He can't draw the whole tree at one time, but he can still decide how he intends to draw it. Actually in practice he could still make a diagram of future history as he intends it to proceed, so in that sense he still can draw the tree all at once. He just can't create the whole history in reality all at once, the plan would have to unfold over time, just as mortals see it.

To take your analogy to computer software: I develop software for a living; I can relate to this. In this case we're looking for the inputs to use to get "good" outputs. So the analogy might be to someone using a mortgage calculator program to analyze a home purchase decision. He puts in the price of the home he's looking at and the interest rate and gets out a monthly payment. He decides that that payment is too high, so he's going to have to settle for a lower-priced home. So he inputs a different home price. He may tinker with this until he finds a combination of home price and monthly payment that he is satisfied with.

This does not lead him into some endless loop of circular logic. The fact that he decided to change the inputs doesn't create an unsolvable self-reference problem, because the user is not part of the program that he is running. He is standing outside feeding inputs. Even if you consider the user to be part of a larger "system" seeking to find an optimal value, there is still no unsolvable self-reference problem. He just plays with values until he gets a satisfactory answer. I don't know if God "plays with" decisions in some "history modelling system" until he gets the desired results, but the concept is the same.

BTW Assuming that God is bound by logic, statements like "God can do anything" should not be taken literally. (When I say that I take the Bible literally, I do not mean that statement literally! The Bible clearly includes poetry, figures of speech, etc. 1 Cor 15:27 says that the word "all" in Psalms 8:6 is not to be taken absolutely literally.) I do not suppose that God can make a triangle with four sides, as that would be logically impossible. You might make such a problem a trick question by redefining the terms in some way, but that's just a trick. By the understood definition of "triangle", it must have exactly three sides. When the Bible says that God is all-powerful, even if taken literally that does not mean that he can do anything. Suppose I told you to draw a four-sided triangle. You reply that this is impossible. So I suggest that you get ten really strong men to help you. Will this make it possible? What if you have the entire output of a nuclear power plant? Clearly more power will not help, because the problem is not lack of power, but logical impossibility.

Sorry for the long answer. Interesting question on which, I am sure, lots more could be said.

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I like your point about nonsense not being overcome by power. It reminded me of a C.S. Lewis Quote: Can a mortal ask questions which God finds unanswerable? Quite easily, I should think. All nonsense questions are unanswerable. How many hours are there in a mile? Is yellow square or round? Probably half the questions we ask — half our great theological and metaphysical problems — are like that. – Steven Apr 11 '12 at 15:58

I agree this is a strawman. You're assuming in step 2 that God would act to stop something from happening because he didn't want it to happen. That pretty much flies in the face of the entire story of scripture and all of human experience. Bad things happen to good people, and sometimes bad people live a good life.

He causes his sun to rise on the evil and the good, and sends rain on the righteous and the unrighteous. Matt 5:45

The entire story from the fall to the crucifixion is an epic saga of things happening in spite of what God wants. Jesus knew that his gift would be destroyed. By your logic he would have chosen not to give it to prevent the rejection, but that isn't what happened.

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God intervenes in happenings a lot. If he didn't it would be deism. Just Jesus visiting a desert, or God killing everyone except one family, or turning women into salt, etc, are examples of God intervening. – danpalmer Apr 10 '12 at 20:16
@danpalmer: that seems logically different to me than the original question – kurosch Apr 10 '12 at 20:29
the original question is 'can these issues be reconciled with God being all powerful and all knowing', and it seems they can't. If God needs to act in the universe as he has done throughout history according to the bible, this problem arises. – danpalmer Apr 10 '12 at 20:53

I think I can show an approach that is quite similar to the halting problem. Here's a thought experiment that my friend once presented to me:

1. Omnipotent being (god 1) creates another omnipotent being (god 2).
2. God 2 may attempt to destroy god 1.
3. God 1 may attempt to prevent its destruction.

There are two possible outcomes:

a) God 2 is omnipotent, so he destroys god 1, so god 1 isn't omnipotent, or

b) God 1 prevents its own destruction, so god 2 isn't able to destroy him, so god 2 is not omnipotent, so god 1 is not omnipotent as he cannot create another omnipotent being (which should be possible for omnipotent being).

I can't think of any flaws in this reasoning and I believe it clearly shows that logic doesn't allow for existence of any all-powerful beings. So either (1) all-powerful God is not constrained by logic, or (2) "being all-powerful" doesn't mean "capable of doing literally anything".

# Scenario 1

In this scenario God is truly all-powerful, ie. capable of doing anything, even if it contradicts logic. The paradox in my friend's reasoning isn't solved, but it doesn't have to be solved. The problem with this scenario is explained in the question:

It is possible to say that God does not conform to logic, however if you believe this you must also accept that, from our point of view, we can then say nothing about his nature, his actions or his existence.

On the other hand, omnipotent God could choose to restrict himself to logic in some aspects, so we could actually talk about his attributes. Basically, this approach relies on the assumption that God's nature is (at least partially) undecidable. The halting problem can be solved for some inputs, but not for all of them.

In a nutshell: Logic operates on `0`s and `1`s, while God operates on real numbers. `0` and `1` are still possible options, though.

# Scenario 2

"All-powerful" shouldn't be taken literally. We need a weaker definition of omnipotence. For example:

God is beyond time and He is all-knowing, so he can avoid running into paradoxes. Considering whether God is capable of creating another god is invalid, because He would never do it. Paradoxes are guaranteed to never happen.

To avoid paradoxes the outcome of any possible choice has to be known. So God has to be at least as powerful (in computational sense) as a non-deterministic Turing machine that can solve halting problem in an instant. This is counter-intuitive and hard to grasp, but doesn't break logic.

Limiting God's power even further leads to some problems. For example to lower computational requirements you could require all input to be deterministic, but that would strip humanity from free will. There's also some minimum level of power that we certainly can attribute to God, like the power to violate the laws of physics (Jesus walking on water).

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"Logic operates on 0s and 1s, while God operates on real numbers. 0 and 1 are still possible options, though." I really don't understand what this means. – Mr. Bultitude Jan 18 '15 at 1:01
@Mr.Bultitude I'll try my best to explain it. In formal logic, statements can be either true or false. In IT they are often represented as `1` and `0`, respectively (why?). Our minds are naturally constrained by logic. For example, anything can either exist or not exist, even if you don't know which is true. I don't know if UFOs exist, but I know for sure that they definitely exist or not exist. Now, if God isn't constrained by logic, then it's possible that we can't grasp some of His attributes. – gronostaj Jan 18 '15 at 10:57
For example he could be all-knowing and not all-knowing at the same time, or partially all-knowing (which, in logic, would simply mean He is not all-knowing, but without logic it means something different). This is what I mean in the real numbers metaphor: there are infinitely many numbers bigger than 1 and smaller than 0, and there are infinitely many real numbers between 1 and 0. The real answer to "Is God all-knowing?" could possibly be, let's say, 1.73 or 3.16 or -5, which doesn't make sense for us. Yet he can make the answer 0 or 1, so that we can understand this part of His nature. – gronostaj Jan 18 '15 at 10:57