Is a table levitating possible, even at a minuscule probability?

[syed]

Just to rephrase to make sure there's no confusion, I'm not counting physical trajectories; I'm counting all the sequences of states you can “freely imagine” if you treat each instant as an independent choice.
For three discrete time-steps and 200 000 possible instantaneous states, you’re constructing (200k) ^ 3 possible sequences. But almost all of those sequences violate the dynamical laws: the state at time t1 has to be the time-evolution of the state at t0, not an arbitrary pick. The same goes for t2.

If you impose the actual equations of motion, the only surviving sequences are exactly those one-to-one trajectories that start from valid initial conditions, so the set collapses back to size 200,000

 

[PeterDonis]

I think you're misreading.

I think you're confused. See below.

I'm talking about imagined trajectories.

What does that even mean? Physics doesn't work by "imagining" things.

suppose that at any instant of time, you can describe the current state of particles in 200,000 different ways.

You can't. There's just one state. You might not know what it is, but there's just one.

What you might be groping towards (but I'm not sure--see further comments at the end of this post) is an epistemic concept of probability--the standard ignorance intepretation, where we assign probabilities to different possible states because we don't know what the actual state is. We have incomplete information, and we use that information to formulate some kind of statistical description of what we know, in order to make predictions as best we can.

In such a statistical description, if the underlying dynamics is chaotic, i.e., has sensitive dependence on initial conditions, then it is true that our ability to predict gets worse and worse as we try to predict further out in time, because of something like an exponential combinatorial explosion of possibilities. However:

Wouldn't most of these imagined trajectories have a literal probability of 0 being realized?

Not in the statistical description I just described, with chaotic underlying dynamics. The possible trajectories are not "imagined"; they're possible as far as we know, because the initial conditions could be the right ones to produce them. So all of them have nonzero probability.

I'm not counting physical trajectories

Then you're not doing physics. and your posts are off topic here, and you need to stop posting along these lines.

I'm counting all the sequences of states you can “freely imagine” if you treat each instant as an independent choice.

This is not physics. We're talking about physics here, where obviously the state at each instant is not independent of the previous states; the dynamical laws relate them.

I'm going to close this thread for moderation at this point, because it seems like your OP question was based on something that has nothing to do with physics. If you think there is still an actual physics question here that hasn't been answered, PM me and we can discuss it.

 

[Dale]

I think this is a poor question. It is not well defined. What counts as levitating? How far off the ground? How long? Are we allowed to use a jet pack? Ropes? EM fields?

This ranges all the way from any day every day to not in a trillion lifetimes of the universe.

So without some serious clarification we are better off counting hairs in our navel. Particularly since I get the impression that you don’t actually care about tables levitating but have some other question that you prefer to dance around.

Just consider that question carefully and then ask it directly.