The universe is incomplete or inconsistent

Great Debates
1

Godel’s Incompleteness Theorem has been expressed informally in any number of ways; I have always liked the notion that anything powerful enough to express at least Peano’s Axioms (that is, arithmetic) will be incomplete (that is, contain undecidable propositions).

The underlying assumption in this post is that it makes sense to apply mathematics/logic to the universe; that is, there is a sense in which the universe can be modeled correctly by mathematics. I am coming right out and saying it as it is important for the point of the post.

My point is this. If any system powerful enough to describe even simple arithmetic can be shown to contain undecidable propositions, and we live in a universe whose model requires at least addition, then the universe is incomplete or inconsistent.

But what does this imply? Does this imply that strict determinism is impossible because there are states which cannot lead to other states (they are undecidable)? Does this mean that strict determinism is somehow possible at the cost of consistency (why waves and particles are the same thing at some level)? Does this mean that there can be no sense in which mathematics “really” applies to the universe because the universe is “obviously” complete and consistent and so must not be describable by math? (I include this for completeness’ sake, though there is no way I would accept that explanation)

Or does it imply none of those things, and is it possible that the universe exists in such a manner that undecidable propositions are expressable in the universe’s math, but they would never actually happen due to the initial state/function of its existence? Do we retain a complete and consistent universe because undecidable propositions never find a real expression? (and is this even possible mathematically speaking)?

I find pondering this to be rather troubling, and would like some input from those with a head that leans toward math/ philosophy/ physics, or just finds the topic to be interesting (of which I am the latter! ;)). Also, please note that I do not require that we have access to this math, only that such a math exists.

2

My completely uninformed, sleep deprived SWAG is that the “laws” of the universe which can be described by mathematics include constraints which prevent “undecideable” situations from occuring. I guess that’s pretty much equivalent to your next to last paragraph.

OTOH, perhaps I am proof that the universe is incomplete and inconsistant. :wink:

3

There’s a very big difference between the universe and any formal system (i.e., mathematics) used to model it. Gödel’s incompleteness theorem (GIT) applies to the later but not the former. In fact, as the universe is not merely a set of axioms, GIT definitely doesn’t apply.

Also, remember that “undecidable” isn’t a truth value. Every proposition in the standard predicate calculus is either true or false. Decidability only describes whether the truth value can be discovered. As I recall, undecidable statements may have counterexamples–they just can’t be finitely expressed.

4

This statement deserves special attention. Remember, the universe is not the model. The fact that our models cant’ be complete and consistent says nothing about the universe; it’s just a limitation on the models.

btw, a theory that just has addition can be consistent and complete. It’s multiplication that screws you up.

5

ultrafilter, bless your heart, I had no idea about the multiplication thing. How very interesting.

You say, “There’s a very big difference between the universe and any formal system (i.e., mathematics) used to model it.”

So you are in the camp that the universe cannot be expressed mathematically? Remember, I place no requirement that we must have or be able to have such a mathematical model, only that the universe can have a mathematic description. God’s math, if you will (for a metaphor).

Because you say, “Remember, the universe is not the model. The fact that our models cant’ be complete and consistent says nothing about the universe; it’s just a limitation on the models.” Then what do the models say about the universe? This seems paradoxical to me when taken to the conclusion. Our idea of models apply to the universe. But our models are flawed intrinsically in that when they are powerful enough to say what we want them to about the universe, they can no longer be both complete and consistent. So either we cannot use math to describe the universe, or the universe is also not both complete and consistent.

IMO this is a whole discussion in itself.

6

I’m leaning that way. Full disclosure: I’m a formalist. I don’t really care what the symbols I manipulate mean, just that there are well-defined rules by which I can manipulate them.

Yep. There is a limit on how well we can mathematically model the universe. I don’t see what’s paradoxical about that.

It’s the same distinction as that between a word and the idea it represents. There are no words to describe some things that everyone experiences, but does that say anything about those experiences? No, it’s a problem of language. Same with math.

And there’s not much room for discussion on the law of the excluded middle in predicate calculus. Remember, the notion of “truth” here is not vague or fuzzy at all. Either there is a sequence of elements in any domain that satisfies a statement, or there isn’t. Therefore, every statement is either true or false.

7

It was only paradoxical when one feels we can use math to understand the world, but that somehow what we understand about the math doesn’t apply to the world. See what I mean?

Also, “Either there is a sequence of elements in any domain that satisfies a statement, or there isn’t.” Right. So which is it for undecidable statements? I am asking a literal question here. If we assume they are true or we assume they are false, what are the consequences? If they can be either without obvious effect, what does that say about LEM?

8

We don’t know, and in fact cannot know within the context of the formal system in question.

The two assumptions each lead to a new formal system, a different formal system in each case. Those two systems themselves have undecidable propositions, which we can assume to be true or false, and that in turn leads to still more formal systems. By continuing in this fashion, we can construct an infinite number of distinct formal systems, each of which contains the “core rules” of number theory but no two of which are equivalent.

It may seem reasonable to assume that one of those formal systems is the “best” approximation to the number-theoretic behaviour of the real world, but even if that is the case, we may never know which formal system is the “truth”. And it is of course possible that none of the possible formal systems are the “best” one; they may, for example, be further and further approximations to an unattainable goal.

But I’m digressing…

Regarding your OP: I honestly don’t think the question is meaningful as it stands. Even if we assume the universe can be expressed mathematically, that’s a far cry from saying the universe is a formal system, and Godel’s theorem is a very specific statement about formal systems of number theory.

In fact, if the universe can be completely expressed mathematically then it would probably be in the form of a dynamical system which is (from a mathematical point of view) a much simpler thing than a formal system. A dynamical system just consists of a parameter space, an initial condition, and some kind of transition function (e.g. a differential equation) which uniquely determines how the system evolves over time. But a formal system is very undeterministic: while there are rules telling you how to prove new statements in terms of old ones, there’s no rule at all telling you which statement you’re supposed to prove next. If there were, math would be a heck of a lot simpler…

9

Conisder the decimial expansion of pi: 3.1415926535… What is the limit as N goes to infinity of the ratio:

(No. of 9’s in the first N digits)/(No. of 8’s in the first N digit)

AFAIK, nobody know how to calcualate this limit or if a limit even exists. Suppose that this limit is indeterminable within the postulates for the real numbers. This would mean nothing, as regards the actual, physical universe, I would guess.

The physics and the math have unrelated problems of decidability.

There must be lots of undecidable mathemetacal statements about the real numbers that simply do not translate into any physical manifistation.

10

Math Geek, “Even if we assume the universe can be expressed mathematically, that’s a far cry from saying the universe is a formal system, and Godel’s theorem is a very specific statement about formal systems of number theory.” Then, for my edification, can you distinguish how the universe differs from a formal system?

“In fact, if the universe can be completely expressed mathematically then it would probably be in the form of a dynamical system which is (from a mathematical point of view) a much simpler thing than a formal system.” The tricky thing about the universe is that it contains formal systems (by inspection). So I doubt that a theoretical model of the universe would be a simple dynamically evolving function (no matter how hideously complicated), and even as such, must still be given in some formal system anyway, wouldn’t it?

What we have here is the fact that the universe is capable of expressing Peano’s Axioms by the virtue of us being in the universe and dreaming them up. I really think this is a metaphysically sticky point. It is even stickier in that we use math to understand the world around us, but yet would try and hide from reverse implication. If the second incompleteness theorem is true (and I see no reason for anyone to deny it) then either there could exist no formal system capable of proving the consistency of the universe, or the universe says it is consistent and is thereby implicated (by the so-called second incompleteness theorem) to be inconsistent.

The other answers left are rejection that the universe contains math, a rejection that math describes the universe, or… well, I don’t know. I’m driven to conclude that if we think math is true in some sense, and that we use it to actually “understand” the universe in some sense, then we are at a point where the universe must be either incomplete or inconsistent.

Consider the position of Berkeley in this debate, if it helps, to see that there is no matter, only mind; or perhaps a strict materialist view, that there is no mind, only matter (if we want to include both and be some kind of dualist then I will have nothing to say, unfortunately, as that seems to needlessly complicate the issue at hand).

…
[further thought on preview]
If we say the universe isn’t a formal system, then doesn’t that alone tell us it isn’t consistent and complete?
…
And to continue with the hijack (a delightful one, IMO), “The two assumptions each lead to a new formal system, a different formal system in each case.” But still, what does this say about the law of the excluded middle? I think it is rather troubling, in some respects.

11

As far as I can tell, nothing presented so far has any bearing on the law of the excluded middle. Perhaps you could explain what you’re thinking?

12

How can I say it… the law of the excluded middle says that logical propositions are either assigned a value of true or false. But undecidable statements may be assigned a value of true or false… or true and false? I’m not sure how to say this eloquently with the implication I have in mind, possibly a sign that I’m talking nonsense.

This is like asking if two parallel lines ever meet. LEM says they either do or they don’t, but in fact we say that they can and they can’t, so where did LEM get us, really? Did it say anything at all about parallel lines? For years and years geometricians wanted to prove that two parallel lines never met because they felt it was not an intuitive axiom. But it is reduced to the level of the axiom; we must assert or refute its truth explicitely because otherwise it is an indecidable proposition in itself, right? So is LEM telling us this is either true or false when it can be both? Or are we saying it isn’t the same LEM in Euclidean geometry that is in non-Euclidean geometry? But isn’t LEM higher up on the chain than the geometric axioms? LEM applies at the most basic logic system, and in principle have not all branches of math been derived from that, with the addition of other axioms? (serious, not rhetorical, question) This being the case, the LEM that applies to Euclidean geometry is, in fact, the same LEM that applies to non-Euclidean geometry. We haven’t changed it, only the additional axioms, at least one of which (and the crucial one, in the case we are discussing) is an axiom because we couldn’t prove it!

Does this help explain my problem?

13

I only have a few moments before I have to go offline for the night, but I wanted to address this:

I think the crux of the issue lies here. There is a difference, as I see it, between the universe “containing formal systems” and the universe containing things which can reason about formal systems. I see no immediate contradiction in the possibility that the universe as a mathematical system is not complicated enough to “contain undecidable propositions” (whatever the heck that’s supposed to mean) while at the same time it contains entities which are capable of reasoning about formal systems which do contain undecidable propositions.

For example, consider Conway’s “Game of Life”. It has been shown that an infinite 2-D cellular automaton following Conway’s rules will be Turing-complete; that is, it can be used to model any computation. In particular, we can use such a hypothetical automaton to model a computer program which blindly searches the space of provable statements in a formal system, using the rules of the system one by one to produce new provably true statements. It follows by Godel’s theorem that there exists statements which the automaton will never be able to prove or disprove, just as there are statements that we will never be able to prove or disprove within the confines of number theory.

However, the existence of such undecidable statements has no bearing on the functioning of the automaton. Our hypothetical 2-D cellular automaton “universe” will continue to run the program forever, perfectly consistent in own rules despite the incompleteness of the formal system about which it is “reasoning”, because the automaton rules and the formal system rules are two seperate entities. In the same way, the existence of undecidable statements of number theory does not necessarily have any physical meaning in our universe.

14

No offense, but I’m gonna have to go with the “you’re talking nonsense” line here. A given undecidable sentence is either true or false, but it’s only one–not both. Undecidability is independent of the truth value of a sentence; it’s a statement about the provability of the sentence. The law of the excluded middle stands.

Parallel lines never meet. That’s a definition, not an axiom. In non-Euclidean geometry, what changes is the number of lines through a fixed point which are parallel to a fixed line. That aside, axioms aren’t undecidable. They are taken to be true for a given system (whatever that word means here). It is not in general true that there is only one line through a fixed point parallel to a fixed line, but it is true that in the Euclidean plane there is only one line through a fixed point parallel to a fixed line.

FWIW, it’s easy to prove that last statement. What the geometers of yesteryear were trying to show is that the statement before it is true, that Euclid’s parallel postulate holds outside of the Euclidean plane.

The law of the excluded middle doesn’t apply here, as far as I can tell. There’s no problem with taking A as an axiom in theory K, and not-A as an axiom in theory H. It’s just that the axioms of theories H and K can’t be true at the same time. Note that H and K could have a large set of theorems in common, though; perhaps removing A or not-A when it occurs leaves the same theory.

Does this make sense?

15

I have spent the past two hours playing with a 30X30 version of life. Christ on a crutch.

16

But this is precisely my problem. Which one is true?

17

Maybe neither. Depends on their other axioms. If the other axioms are true, then it depends on the truth of A. If A is true, the one with A as an axiom is the true theory; otherwise, it’s the other one.

Note that if the axioms of K are false, and B is a theorem of K, then the statement “{axioms of K} -> B” is still true.

And yeah, life is pretty cool. The math of it is wicked, too–Conway’s a genius, IMO.

18

By the logical converse of Godel’s Incompleteness Theorem, if the universe is not a formal system, then it might be possible that it is both consistent and complete.

It kind of seems like we’re asking the universe to be a computer that embodies abstract rules of logic and verifies whether statements are true or false. Maybe that’s not as wacky as I thought at first.

You could look at the scientific method that way:

  1. I hypothesize ‘If P, then Q’
  2. I set up an experiment where I manipulate a part of the universe to comply with the condition P
  3. If Q occurs, then the universe system has verified my hypothesis.

The drawback is that I have only proved it for one instance. We need to use induction to prove our hypotheses when P is a set of large or infinite number of situations. (And induction in this case is not the rigorous mathematical version.)

What if your hypothesis ‘if P, then Q’ contained a P that was impossible to set up in the universe? Then your universe/computer would be unable to test your hypothesis. Therefore incomplete.

19

But it can’t be “maybe neither”, ultrafilter. Either P or ~P, right? That is, P V ~P is always true. That’s what LEM is telling us, and that is my problem with it. For all propositions P, either P or ~P is the case. That seems like a tremendous leap to me. Are there many proofs that rely on the fact that LEM exists. that is, where only one side of the truth value can be proven?

Math Geek

I don’t see how, other than adding a middleman in the picture. Instead of the straight shot from Universe as container for formal systems, we have universe as container for humans which are containers for formal systems. :shrug: I think that complicates the issue unnecessarily.

Let me restate my problem for clarity[list=1][li]The operation of the universe may be abstracted to the level of a formal system[]This formal system is a subject of GITs.[]Thus, the universe is incomplete or inconsistent[/list=1][/li][quote]
However, the existence of such undecidable statements has no bearing on the functioning of the automaton.
[/quote]
Au contraire, it has a tremendous effect. If its actions are successive proofs of more and more theorems, then the actions it is forbidden to do are theorems which cannot be proved. (assuming that all the discoverable theorems are countable, of course… do you know if anyone has shown this to be the case?)

I don’t think that is the case.

heresiarch

Is that really the case? Don’t “consistent” and “complete” only apply to formal systems to begin with (from the context of Godel’s theorem)?

Precisely! Sort of Douglas Adams-esque here, in one sense, though not the sense I intend most strongly, though that depends on how one views metaphysical reality to begin with.

Okay, maybe ‘precisely’ was a bit of a hyperbole. In fact it is close, though, to my original intention. We could very well ask ourselves if the universe is Turing-complete, in which case it would have to be incomplete by GIT, and by way of there are things that should be able to happen but cannot (states that can never be the case, even though nothing forbids them). This is a very strange situation.

But because the universe is a container for Turing complete “things” (as complete as they can be since there is no infinte memory to work with), then we already know that it is Turing-complete. That’s like saying that the game of life here-mentioned is Turing complete but the computer on which it is implimented isn’t… seems downright false, doesn’t it?

Indeed.

20

Huh? You have two theories, one of which has A as an axiom, and one of which has ~A as an axiom. With no restrictions on their other axioms, why can’t they both be false? Perhaps they both contain B & ~B as an axiom.

**

There are only a countable number of finite-length sentences, therefore there are only a countable number of finite-length theorems. btw, in order to be a theorem, a sentence has to have a proof. “theorems which cannot be proved” is a contradiction in terms.

But Math Geek’s automaton can do exactly what it was intended to: it can discover all the theorems of a theory. It may not find all the sentences that are true in that theory, but that’s not a problem for a theorem-prover.
**

The game of life is only Turing-complete when played on an infinite grid. If you’ve only got a finite amount of memory, then you’re only playing a finite version of the game.