Incompleteness and Uncertainty

[fruitfly]

Hello,

I'm new to this forum so I apologize if this has already been discussed.

While reading Godel's incompleteness theorem, the possibility occurred to me of interpreting Heisenberg's uncertainty principle in terms of an underlying universal axiomatic system. The basic idea being that any part of the universe behaves according to a fixed set of laws or "axioms" and that at any given point in space/time all events that do not contradict these laws are possible. Therefore, if the underlying axiomatic construct is robust enough there are events that can occur but which do not arise from any possible sequence of axiomatic laws applied to a previous cause. I'm wondering if there are any current physicist that subscribe to this interpretation? Thanks.

 

[Bob3141592]

I'm not a physicist, but this is my take on the topic. Godel's Theorem is specifically about the nature of formal systems, that is, a finite set of explicit rules about the manipulations of symbols given a finite set of defining axioms. Formal systems don't have meaning beyond the system itself.

In physics, it may be reasonable to assume (but it is certainly not necessary) that there are only a finite number of laws. It is also natural to think that physical objects themselves are equivalent to the symbols that describe their behavior, but again, this is a matter of philosophy or theology, not math or physics.

What Godel's Theorem says (I believe) is that using the operators of the formal system, statements of that system can be derived that are either inconsistent (can be shown by the system to be both true and false, like the continuum hypothesis/axiom of choice), or that the system is incomplete (there are true statements of the formal ystem that cannot be derived by the operators).

The HUP is a description of a property of matter. In that regard, it is no different from Newton's or Maxwell's laws. Godel's Theorem applies to the HUP no more nor no less than it does to any equation in physics.

And what the HUP says (I believe) is that matter can only be described by operators that do not commute. For example, using the most commonly paired attributes of position and momentum, p q - q p >= ih/2 pi. There are other pairs, and I think all attributes of matter are paired with some other attribute, like energy and time, or charge and phase. The HUP does not say that physics is incomplete, and it certainly doesn't say that physics is inconsistent.

Is the universe a formal system? Not from our perspective, at least. Once you think that an object like a car or a person is a real thing that means something, you have already gone beyond the domain of Godel's Theorem. But note that just because Godel's theorem doesn't apply to real objects, that doesn't mean we can completely understand those objects, or that those objects are necessarily consistent. Especially if that object is a woman. :-)

 

[selfAdjoint]

What Godel's Theorem says (I believe) is that using the operators of the formal system, statements of that system can be derived that are either inconsistent (can be shown by the system to be both true and false, like the continuum hypothesis/axiom of choice), or that the system is incomplete (there are true statements of the formal ystem that cannot be derived by the operators).

What Goedel says is that if you have a formal system that is capable of proving arithmetic, then you can map it into itself using arithmetic operations for prooof techniques and prove that there are statements which can be both proved and disproved in it. And you can't eliminate this problem by addng new axioms, because new undecidable propositions will always arise. A formal system like this is called incomplete. In particular, the various axiomatizations of set theory are incomplete.

However geometry, and by extension real variable theory, apart from any set theory sections, is complete. This is Tarski's theorem, which ought to be as famous as Goedel's theorem, but isn't.

 

[Bob3141592]

However geometry, and by extension real variable theory, apart from any set theory sections, is complete. This is Tarski's theorem, which ought to be as famous as Goedel's theorem, but isn't.

That's interesting. Sorry if this is more appropriate for the math forum, but I'm curious. Isn't geometry also a formal system, with it's axioms and rules of operation? Isn't geometry equivalent to algebra in it's ability to handle arithmetic? What does geometry have (or lack) that exempts it from Godel's Theorem?

 

[Hurkyl]

What does geometry have (or lack) that exempts it from Godel's Theorem?

One thing in particular is that geometry lacks (and so does the algebra of the reals) the ability to define the term "integer".

Since geometry can't talk about integers, it certainly cannot talk about theorems involving integers; geometry does not contain number theory, and thus Godel's theorem does not apply.