@irecorsan That is a very good point, thank you. And presumably, in your parlance, I can say the probability space does not include flying pigs.

@MarcoOcram I think Bumble is on more solid ground when he mentions that if a probability is exactly 0 (or 1), bayesian updates will be unable to move it. That seems undesirable, so one might as well assign an epsilon probability to events that one doesn't want to exclude - at least from a bayesian point of view.

@frank, point taken, although the probability of finding some evidence that might make you update the probability of a flying pig is itself zero in my book!

@MarcoOcram - sure - I just thought that Bumble's point on how bayesian will be inop of 0 or 1 priors was relevant. Flying pigs are a long way off! :-)

It bugs me when people use "infinitesimally small" and "zero" interchangeably, because this conflates literal impossibilities with infinitesimally unlikely events that still happen. (Although this may be an accepted practice in mathematics.)

@NotThatGuy I think that's because infinitesimally small is not an entity, just like infinity is not an entity. You can't do arithmetic with infinity, so the probability of a specific value in a continuous random variables is indeed zero, not something that is infinitesimally small. CMIIW

@justhalf Infinitesimals do not exist in the standard real number system, but they are definable in other number systems and many mathematicians use them. So NotThatGuy is justified in complaining about conflating infinitesimally small numbers with zero. Also, in measure theory we can speak of a quantity being 'measure zero' which is more precise than just saying zero.

I guess you have a point, yes. Is the probability of a specific value in a continuous random variable something that has a measure zero?

@justhalf You can have countably infinitely many points of non-zero probability within the domain of a continuous random variable. And the remaining uncountably many possible outcomes of zero probability can still have a non-zero probability as a whole.

There is a non-zero chance of a pig being born with functional wings, or for a an abrupt vacuum to suck it arbitrarily high into the air, or for it to quantum-tunnel into the sky.

Excuse me for asking, but what do continuous random variables have to do with OP's question?

Besides, how do you know that flying pigs is impossible and simply not so extremely unlikely that it has never happened? If you can't prove it, I think you're wrong.

@HelloGoodbye regarding your first question above, please see the answer below from Sanjevo. Regarding your second, if you think pigs can fly then pigs can fly.

@BlueRaja-DannyPflughoeft the saying 'pigs might fly' refers to pigs as we know them. If a pig were born with functional wings then it would not be a pig. As for vacuums and quantum tunnelling and the like, they too do not satisfy the condition 'if you take the saying at face value' which I specified in my answer.

but what is the probability of ignoring planes and hurricanes?

@HelloGoodbye The answer says zero-probability does not imply impossibility, which is the same as saying possibility does not imply non-zero probability, which is what the OP asked for. Maybe that's what confused you.