Is there some connection between incompleteness theorem and uncertainty principle?

Question

I understand close to nothing about the incompleteness theorem but it sounded somewhat like the uncertainty principle to me even though I didn't understand anything. Are they related?

Answer 1

No, these really have nothing to do with each other.

In quantum mechanics, observables (things that can be measured) are represented by operators that act on the wave function. For each operator, there are a special set of wave functions called eigenstates that will always give the same value for the corresponding observable when measured. For any other wave function, the measured value will have some probability distribution over the possible values for the measurement. Thus, we see that only eigenstates have a certain value for the observable. All the rest have some degree of uncertainty.

In order to measure two quantities with certainty simultaneously, there must be a set of wave functions that are eigenstates of both operators. This is only possible if the two operators commute, and this is not always the case. The best-known examples are the position operator ($x$) and the momentum operator ($-i\hbar \partial_x$). For non-commuting operators, the eigenstates of one operator are the maximum-uncertainty states of the other. For example, the eigenstates of the momentum operator are $e^{ipx/\hbar}$, which have their probability density evenly distributed across all space. Because of this, there's a trade-off of accuracy in one observable vs accuracy in the other, and the uncertainty principle quantifies the limits of this tradeoff.

Note that this is a statement about the physical world according to the model of quantum mechanics, not about the limits of knowledge. An electron (for example) having a position and momentum with no variance is simply impossible, much as going faster than light is impossible in special relativity.

Godel's incompleteness theorem, on the other hand, deals with proof theory. In particular, it shows that any consistent and sufficiently complex axiomatic system is able to formulate statements that cannot be deduced from those axioms. It turns out that the bar for "sufficiently complex" is quite low: even basic arithmetic qualifies. Now this is a little out of my field of expertise, so take this with a grain of salt, but it does seem to mean we'll never have a set of axioms that captures all of mathematics. Which is rather unfortunate. Seems like that'd be a nice thing to have. On the other hand, we do seem to be able to get "enough" of it, in a sense, unless the continuum hypothesis really keeps you up at night.