Say that there is a Theory of Everything (ToE) and that this world is its model. — alcontali

Shouldn't it be that a ToE is a model of the world rather than the other way around?

I would say if we have free will then we have the freedom to create alternative worlds, which doesn't necessarily imply that there are alternative worlds, but they can be created.

Shouldn't it be that a ToE is a model of the world rather than the other way around? — leo

Well, in normal English, yes. In model theory, no. In model theory, a theory T is a set of rules, while a model M is a set of values, i.e. an "interpretation", that satisfy these rules.

The universe itself is not a set of rules. It is a set of values that satisfy the rules of the ToE.

Then the fact, that there are true facts in the universe (model), that cannot be proven from the ToE (theory), implies that there are other such models.

Well, in normal English, yes. In model theory, no. In model theory, a theory T is a set of rules, while a model M is a set of values, i.e. an "interpretation", that satisfy these rules.

The universe itself is not a set of rules. It is a set of values that satisfy the rules of the ToE. — alcontali

Okay, but then it seems you assume that this set of rules existed prior to the universe which would be an instantiation of these rules, whereas we have no evidence of these rules existing before, rather we attempt to infer a ToE from the universe we do observe.

There are many universes consistent with an incomplete theory, but even if somehow we found a ToE, and even if somehow we knew that the rules of the ToE existed before the universe, it still wouldn't prove that there are other universes, because it seems to me the incompleteness of the theory only implies that many possible universes are consistent with the theory, not that these universes exist as more than possibilities.

However if we have free will it seems to directly imply that we can create other worlds. As to why we seemingly can't, it could be that our will is constrained by some other will, in which case our will is not completely free, or it could be that we willed to give ourselves the appearance that it isn't free even though it is, and then in principle it could be possible to get out of that state and apply our free will and we just haven't found out how yet.

Also it seems a will cannot be completely free as long as there are other wills, unless there is only one will and we are all a part of it and the one will gave us a limited will, which is the view of a version of idealism in which the whole universe and ourselves exist within a cosmic consciousness.

Personally I hold the view that wills are not constrained by rules or laws but by other wills, or more specifically as I mentioned in another thread that there are two wills constraining one another, the will of Good that seeks to unite and the will of Evil that seeks to separate, that we are part Good part Evil and so our resulting will is constrained by the conflict between the two, and that in order to have free will we would have to make the will of Evil disappear, and then we would all be united as one, with a free will that would seek to do no evil, for having a free will is having the ability to do anything one wants, rather than actually doing anything, even if one could in principle.

Okay, but then it seems you assume that this set of rules existed prior to the universe which would be an instantiation of these rules, whereas we have no evidence of these rules existing before, rather we attempt to infer a ToE from the universe we do observe. — leo

These rules did not have to exist before.

If you see a model M, then you can argue that they satisfy a theory T. Fine, but now you also see that there are facts in the model that cannot possibly be predicted by such T. That creates the situation that there are statements that are true in model M but not provable in theory T. According to model theory, this means that there must be at least one other model M' which also satisfies theory T but in which these facts are false. That is why these facts are true in M but not provable in T.

The term "free will" in this context means: there are facts in this universe for which no theory about that universe could ever predict them. Hence, according to model theory, there are other universes that satisfy the theory that governs our universe.

There are many universes consistent with an incomplete theory, but even if somehow we found a ToE, and even if somehow we knew that the rules of the ToE existed before the universe, it still wouldn't prove that there are other universes, because it seems to me the incompleteness of the theory only implies that many possible universes are consistent with the theory, not that these universes exist as more than possibilities. — leo

According to model theory, if there are no other models that satisfy the theory, then all true facts in the single model will be provable/predictable from that theory. As soon as you detect a true fact in that model, however, that cannot possibly be predicted from its theory, then there must exist other models. Otherwise, such true but unprovable facts cannot exist.

The theory of our universe is consistent, because otherwise it cannot have a model, i.e. our universe, which clearly exists. If free will exists, then our universe has true facts that are unprovable by any theory about it. Therefore, there must be other models, i.e. other universes.

However if we have free will it seems to directly imply that we can create other worlds. — leo

We cannot create these other worlds.

However, without the existence of these other worlds, we cannot have free will. Our free will produces true facts in this universe that cannot possibly be proven in any theory about this universe. This is only possible, if there are other universes that satisfy the same theory as the one which our universe satisfies.

It is the same for natural numbers. Gödel proves that there are true facts about natural numbers that are not provable from the theory of arithmetic. Therefore, there must exist carbon copies of these natural numbers, i.e. alternative models, in which these facts are false. The existence of these nonstandard models of arithmetic is not optional. A true fact that is not provable means that the same fact is false in another model.

Hi alicontali,

If I am understanding it correctly, your argument takes this form:
1. There is a Theory of Everything (ToE).
2. This world is the ToE’s model.
3. If we have free will, then the ToE is not able to calculate what we will be doing.
4. What we do is true in our universe but not provable from the ToE.
5. The existence of true but not provable statements in the ToE means that the ToE is incomplete.
6. Incompleteness of a consistent theory automatically implies the existence of more than one model.
7. Therefore, if free will exists, then alternative worlds (that we may call heaven and hell) also necessarily exist.

Although I think this argument is valid, I find the conclusion and its implications hard to agree with. I think that if free will exists, it doesn’t mean that heaven and hell are the alternative models. The alternative models to this world could be anything. They could be exact replicas of this world with one small detail that doesn’t match up. I don’t think it is safe to assume from the fact that we have been given free will that the other models are just where we go after we die.

Basically, I don’t think this argument warrants the conclusion that heaven and hell exist. I definitely believe that the ToE being incomplete warrants other models of this world, but I would say something more along the lines of the multi-universe hypothesis. I would conclude that God has made other models, but they are not heaven and hell. I think they are more so just other universes like ours with slight variations.

I think your argument is a really good way to prove the multiple universes hypothesis which in turn helps to prove God’s existence. If you are not familiar with it, check it out and see if what I am saying makes sense. Although heaven and hell may be the other models, I think it’s more likely that some other worlds less specifically characterized are the other models.

If I am understanding it correctly, your argument takes this form:
1. There is a Theory of Everything (ToE).
2. This world is the ToE’s model.
3. If we have free will, then the ToE is not able to calculate what we will be doing.
4. What we do is true in our universe but not provable from the ToE.
5. The existence of true but not provable statements in the ToE means that the ToE is incomplete.
6. Incompleteness of a consistent theory automatically implies the existence of more than one model.
7. Therefore, if free will exists, then alternative worlds (that we may call heaven and hell) also necessarily exist. — Marissa

Yes, and you actually expressed it more clearly than me. I have a lot of trouble saying this succinctly, and I rarely feel that the idea came across correctly, but this time, I actually do. This is indeed what I meant to say. Thanks!

In the meanwhile, I also discovered Tegmark's ultimate ensemble theory, which seems to say something similar. However, I probably disagree with Tegmark's view on the effect of Gödel's incompleteness theorems. He says we can simplify the universe to a "Computable Universe Hypothesis (CUH) which only includes mathematical structures that are simple enough that Gödel's theorem does not require them to contain any undecidable or uncomputable theorems."

This is pretty much the opposite of what I argue. It is the existence of undecidable statements that leads to the idea of multiple universes. So, simplifying them away would go against that.

I think that if free will exists, it doesn’t mean that heaven and hell are the alternative models. The alternative models to this world could be anything. — Marissa

Agreed.

The actual characteristics of these alternative universes do not result from Gödel's work. His work does not go as far as describing the complete difference between the various models. He just detects that they must be there.

I don’t think it is safe to assume from the fact that we have been given free will that the other models are just where we go after we die. — Marissa

Agreed.

This is another proposition that goes far beyong Gödel's work. It is indeed not possible to reach any conclusion about this just with model theory. The tool is much more limited than that ...

Basically, I don’t think this argument warrants the conclusion that heaven and hell exist. I definitely believe that the ToE being incomplete warrants other models of this world, but I would say something more along the lines of the multi-universe hypothesis. I would conclude that God has made other models, but they are not heaven and hell. I think they are more so just other universes like ours with slight variations. — Marissa

Agreed.

From model theory, there is indeed no answer to this remark. The heaven and hell concept would fit into the this multi-universe hypothesis, but any other system of universes would too. The limitations of model theory do not allow us to conclude anything else, merely by using model theory.

I think your argument is a really good way to prove the multiple universes hypothesis which in turn helps to prove God’s existence. — Marissa

I think that model theory does indeed not help directly proving that God exists. I think that, at most, it could prove that multiple universes exist.

I actually do not even feel the need to prove that God exists. As far as I am concerned, you believe it or you don't. Personally, I instinctively believe that God exists, and that suits me absolutely fine. Seriously, I am not looking for a proof that God exists, simply, because it is of little use to me. There are questions on which I completely trust my instincts and where formal arguments would not even sway me.

Furthermore, model theory is just a tool. It allows us to pick a few otherwise seemingly unimportant structural details from within the universe and draw surprising conclusions about them.

It is a similar exercises as looking around you on earth and picking a few innocuous details from which you can decisively conclude that the earth must be a globe, and that it cannot possibly be flat:

• Watch a ship sail off to sea
• Climb a tree
• Watch a sunset
• Measure shadows across the country

Once you are aware of the hypothesis that there is more than one world, there are most likely clues that you can see from within our world. These details will undoubtedly be innocuous. Otherwise, people would already have picked them up. I consider model theory to be one such clue, but there could be more.

If we have free will, then the ToE is not able to calculate what we will be doing. — alcontali

You are assuming an incompatibilist conception of free will. On a compatibilist conception, free will has no implications on determinism of lack thereof.

If free will exists, then alternative worlds (that we may call heaven and hell), also necessarily exist. — alcontali

Proving the existence of multiple worlds is far removed from proving the existence of anything we might want to call heaven and hell. Lots of people are modal realists without believing the other possible worlds besides the actual world are in any way religiously significant.

If you see a model M, then you can argue that they satisfy a theory T. Fine, but now you also see that there are facts in the model that cannot possibly be predicted by such T. That creates the situation that there are statements that are true in model M but not provable in theory T. According to model theory, this means that there must be at least one other model M' which also satisfies theory T but in which these facts are false. That is why these facts are true in M but not provable in T. — alcontali

Do you agree that saying there are two universes that satisfy theory T does not necessarily imply that the two universes actually exist? That is my point.

For instance there could be a finite universe and an infinite universe that both satisfy a theory T, that doesn’t imply that both universes actually exist, it could be there is only one of them.

Saying that something exists in the realm of mathematics does not imply that it is actualized in reality. I understand your argument, if we know that our universe satisfies a theory T and we have free will then we can conceptualize other universes that satisfy this theory T, but there is a missing step between existing as a concept and existing in reality. Unless you assume that everything that we can think exists beyond our thoughts.

We cannot create these other worlds. — alcontali

Then what you call free will is not really free, it is constrained. And if you define free will as the ability to do things that cannot be predicted, in principle it could be possible to do things that can’t be predicted without having control over them, in which case it would decidedly not be a free will in the way that it is usually considered.

But regardless of the question of free will, I generally think that proving that several concepts (models) are consistent with another concept (a theory) does not imply that these concepts exist as more than concepts. Unless again you assume that everything that can be thought exists beyond thoughts.

Although I think this argument is valid, I find the conclusion and its implications hard to agree with. I think that if free will exists, it doesn’t mean that heaven and hell are the alternative models. — Marissa

Of course not, but if I am understanding it right, then free will consists of an extra input from another world analogous to a player's input into a computer game. The mere existence of a world where I misspelled analogous in this post would not suffice. So we could reasonably call it 'the spirit world'.

But regardless of the question of free will, I generally think that proving that several concepts (models) are consistent with another concept (a theory) does not imply that these concepts exist as more than concepts. — leo

I think in crude terms the argument is that if the existing world is computational and there exists free will, then there is another existing world that influences the existing world.

Do you agree that saying there are two universes that satisfy theory T does not necessarily imply that the two universes actually exist? That is my point. — leo

Well, everything that is provable about one universe in the theory will also be provable about the other universe.

For example, if it is provable from the theory that our universe exists, then it will be automatically provable that the other universe also exists. Another example would be the existence of a particular galaxy. If the theory proves that it exists in our universe, it also proves that it exists in the other universe. It will be exactly the same proof.

For instance there could be a finite universe and an infinite universe that both satisfy a theory T, that doesn’t imply that both universes actually exist, it could be there is only one of them. — leo

These universes will be "largely" isomorphic. For models of natural numbers:

The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

In my impression, a finite universe and an infinite one cannot be isomorphic. But then again, models do not need to be perfectly isomorphic either. It is more about satisfying Skolem's construction constraints (which I haven't investigated in detail yet), mutatis mutandis. Not all other possible universes are sufficiently isomorphic with ours to be models of the same theory, just like not all sequences of numbers are sufficiently isomorphic with the set of natural numbers to be a model for arithmetic theory.

What seems to be needed, to satisfy Skolem's requirements, is that a carbon copy of our entire universe is contained in the other universe, or the other way around.

Our universe is finite, and our time is finite, because it progresses. If our time were infinite, it would not be able to progress by addition of time slices. If the other universe is infinite, its time will also be infinite, and therefore, will stand still. It is pretty much impossible that such two universes (a finite versus an infinite one) would be isomorphic in any way.

Saying that something exists in the realm of mathematics does not imply that it is actualized in reality. — leo

Agreed, unless it is about the ToE. If a proposition is provable in the ToE, then it will necessarily be true in our reality. We do not have a copy of the ToE. Therefore, we cannot use it to prove facts in our world. What I am doing here, is working the other way around: There are facts that even the ToE cannot prove, but that are nonetheless true in our reality. Hence, these facts must be false in nonstandard models of the ToE (In that case, we accept our universe as the standard model for the ToE). Hence, these nonstandard models must exist. Furthermore, everything that is provable in the ToE about our universe, is not only true in our universe but also in all these other nonstandard universes.

I understand your argument, if we know that our universe satisfies a theory T and we have free will then we can conceptualize other universes that satisfy this theory T, but there is a missing step between existing as a concept and existing in reality. Unless you assume that everything that we can think exists beyond our thoughts. — leo

The ToE has special status. The ToE can prove everything that is provable about our universe. Every provable theorem in the ToE corresponds to a true fact in our universe. Hence, if you can prove a claim from the ToE, then you can simply see it in our universe. It will be undeniably true.

Other theories are merely Platonic abstractions that do not necessarily correspond with the reality of our universe. In fact, there is no reason to believe that they correspond with the real, physical world. We use empirical methods (by experimentally testing them) to infuse some correspondence in such theories. This is not necessary for the ToE, because the ToE automatically corresponds with our reality.