Godel's original proof does not use full omega consistency. He only needs a small fragment, namely that if the axiom system proves a program P halts, then P actually halts. This is sigma-0-1 soundness.

This answer is not accurate--- the stochastic process, (on every step after the first (ignoring collisions) is entirely deterministic, and takes pure states to pure states.

Fair enough--- but the usual way mathematics is done is by quoting and using previously proven theorems, and the mathematical work of the physicists generally does not quote previous theorems, but instead constructs the objects in question from scratch. So I think it is in the spirit of the question. The Dirac and Virasoro examples might be inappropriate, I don't know the history of the things very well.

Also, keep in mind that measurements can be pushed up to the level of psychology, the measurement can be thought to happen only at the last step, when you look at the measuring device. Then one can ask what physical reason allows a strong laser tuned to 0-2 transitions to prevent 1-0 transitions. The reason is that transition from 1-0 is accompanied by high frequency amplitudes for transitions to 2, which are entangled with incoming photon phases and therefore incoherent, so the down transition amplitudes get scrambled. The ground state in the presence of the photons is not stationary.

It took me a little thinking to understand why people are saying that quantum zeno is limited by time-energy uncertainty, because it isn't at all. The wrong idea is that a measurement of energy to accuracy "delta" will have to take 1/delta seconds, so that the density of measurements can never exceed 1/delta. The first clause it true--- a measurement will have to take a long time, but the measurements can overlap, so that the density is unlimited.

A measurement can be a failure of interaction. If you have an atom in its first excited state, it will decay to the ground state normally. You can apply an arbitrarily strong laser whose frequency is tuned to the energy difference between the ground state and the second excited state, and this measures when the atom transitioned to the ground state. This will prevent the atom from ever making the transition to the ground state, and there is no theoretical limit to the density of measurements from the uncertainty principle, because the different measurements are free photons.

@Anthony: That's puzzling. As I understood the problem, it is saying that the probability that an n-step walk with a given generator set lands in a bounded set of points decays exponentially. This is false in Z^2 because the Gaussian kernel decays as a power, but it is true in a free group on two generators because you have exponential growth. It is possible that the OP meant an arbitrary subset, but then you could take A to be the whole group minus a few points, and it is trivially false. I think that is overly uncharitable interpretation, but perhaps the question could be clarified.

The question is wrongly worded, but the process might be meaningful: consider a random walk which is restricted by the condition that at each collision it reflects off the rest of the walk so as to stay outside the closed loop formed. This process is instantaneously measure zero as compared to an ordinary random walk (by scaling or by 0-1 laws), but does it define a universality class different than the usual self avoiding walk? Can't you take an ordinary Brownian path and discontinuously reparametrize it so that it becomes a reflecting path?

What are the conditions on the group? I assume free, because your result is true in a free group with two generators, and false for $Z^2$.

There is no direct analog because for ordinary random walk, you don't need to know the past to go forward in time. To go forward in time in a self-avoiding walk, you need to know the past, so you know what to avoid. This means that there is no differential equation form of the process, which is what I think you wanted.

This question is not well phrased--- what you can ask is "what is the effect of measuring an operator again and again, when the operator varies continuously in time".

Considering the little Aristotle I have read, I find it hard to believe he had anything worthwhile to say about modal logic. It is not sufficient to state that some things are "necessary" and others are "possible", many people have done that, but one must also define at least a rudimentary set of formal rules for manipulating negations and conjunctions of these statements.

It is dubious to attribute this to Archimedes--- it has the flavor of European 17-19th century puzzle mathematics, and it's attribution to Archimedes is folklore.

If f(f(x))=exp(x) then f(x)=log(f(exp(x)), so it is not reasonable to take f any more analytic than log.

I assumed when I did the simulation (many years ago, and I no longer have the code) that the number of self avoiding walks of length N should be 4^N times exp(-S), that is, the number of walks reduced by the probability cost of self avoidance. This gave the wrong answer, and at the time, I convinced myself that this was due to some measure subtlety, but I forget what.

A better way to do the simulation is as follows: simulate the path, whenever the path closes a loop, there are always extensions which are outside (lattice connected to infinity), but there might be extensions which are inside too. Whenever you reach such a point, use an inside/outside algorithm (draw an irrational line and count the parity of the number of intersections with the curve) to add a factor of log(number of inside points)-log(number of neighbors) to a variable S. The variable S is eventually linear, and it is the log of the asymptotic fraction of RWs which are not trapped per step.