Case A is that the universe is finite and that cows spontaneously turn into chickens with probability epsilon. Case B is that the universe is infinite and epsilon of all cows actually turn into chickens. In both Case A and Case B your experiences are exactly the same. Your expectation of observing a cow turning into a chicken is the same.
Not quite true... This only follows in Case B if you assume a principle of mediocrity i.e. that you are a "typical" observer. But that assumption leads to known problems in an infinite universe e.g. it implies a strong form of Doomsday argument, where only a tiny fraction of civilisations survive and become space-colonizing. (Otherwise, you'd expect to be part of a long-lived, space-colonizing civilisation of huge population, rather than still on the planet of origin of your civilisation with only a few billion population).
If you DON'T assume a principle of mediocrity, then Case B makes extremely weak predictions about what you should observe : you might, for all you know, be part of the epsilon who observe weird things at some point in their lives.
But that assumption leads to known problems in an infinite universe e.g. it implies a strong form of Doomsday argument, where only a tiny fraction of civilisations survive and become space-colonizing.
What is the problem with this argument, besides an unpleasant conclusion? We haven't actually seen any space-colonizing civilizations, and we would expect to see at least some unless we were either one of the first civilizations in our light cone or if most civilizations weren't spacefaring (possibly because they don't survive that long).
Hi gang,
for the last several months, I've intermittently been wondering about a curious fact I learned.
You see, I was under the impression that the universe (as opposed to just the currently observable universe or our Hubble volume) must be finite in its spa[t|c]ial dimensions. I figured that starting with a finite area of space, expanding with a finite (even if accelerating) expansion rate, could only yield a finitely sized volume of space (from any reference frame), a fraction of which constitutes our little Hubble bubble.
Turns out - honi soit qui mal y pense - my layman's understanding was wrong: "This [WMAP data] suggests that the Universe is infinite in extent (...)"
Now, most (non-computer-scientist) people I've bothered with that answered along the lines of "well, it's really big alright? (geez)".
However, going from any finite amount of matter/energy to an actual infinite amount (even when looking at just e.g. baryonic matter from the infinite amount of galaxies) still seems like a game-changer for all sorts of contemplations:
For example, any event with any non-zero probability of happening, no matter how large the negative exponent, would be assured of actually happening an infinite amount of times somewhere in the our very own universe (follows straightforwardly from induction over the law of large numbers).
Such as a planet turning into a giant petunia for a moment, before turning back.
The universe being infinite doesn't make that event any more likely in our observable universe, of course, but would the knowledge that given our laws of physics, there is an infinite amount of Hubble spaces governed by any sorts of "weird" occurrences - e.g. ruled by your evil twin brother - trouble you? Do we need to qualify "there probably is no Christian-type/FSM god" with "... in our Hubble volume. Elsewhere, yes."?
The difference, if you allow me a final rephrase, would be in going from a MWI-style "there may be another version - if the MWI interpretation is correct - that I cannot causally interact with" to a "in our own universe, just separated by space, there is an infinite amount of actual planets turning into actual petuniae (albeit all of which I also cannot interact with)".