(This evening, shortly after reading this, I had dinner at a restaurant in NYC — and there was Mayor Bloomberg at the next table. I heard someone say, "Nothing's impossible after all.")

I'm not sure what an infinitely small probability would be. Perhaps just a probability of 0? But that sounds like an impossible event. So perhaps you're asking whether all events have some finite non-zero probability of occurring — and whether the events we call "impossible" really just have a very small finite probability.

Philosophers have spent a lot of time trying to figure out what we're actually saying when we assign a probability to an event. Are we making some claim about the world? Or are we making a claim about our degree of confidence in some judgment about the world? I won't go into that here and instead will say a few words about impossibility.

Philosophers often distinguish between different kinds of impossibilities. Some situations would conflict with the laws of logic: for instance, the state of affairs in which I am over thirty years old and not over thirty years old is one that conflicts with the law of logic that says that "A and not-A" is false for every statement A. We might say that that state of affairs is logically impossible, or impossible relative to the laws of logic. By contrast, some situations conflict only with the laws of physics: for instance the state of affairs in which I am moving faster than the speed of light is not a possible one according to contemporary physics. It's one that is logically but not physically possible, one that is impossible relative to the laws of physics. Likewise, we might have situations that we would describe as impossible relative to the laws of chemistry, and so on. And perhaps, when someone suggests that your spouse is having an affair you will find yourself exclaiming that that's impossible, meaning not that such perfidy is inconsistent with the laws of logic or physics, etc., but that it's incompatible with what you believe to be true about your spouse.

If this is the right way to think about impossibility, then nothing is impossible — tout court. A situation is possible or impossible only relative to certain assumptions. And relative to any given body of assumptions, many situations will be impossible.