How about a photon? I don’t need it to locate in a Zeno’s paradox of small space smaller than a Planck length to have it propagate in all directions at the speed of light. I think that your desire to locate an atom in an uncountably infinite number of places is a geometric concept and not a physical possibility. I don’t understand quantum physics, but it is my understanding that atomic and sub-atomic particles do not have a precise location or velocity.
Given the existence of combinatorial explosions, I’m guessing that a bizarro Earth would be an analogue of “2”: it doesn’t exist.
As a demonstration of combinatorial explosions, consider the Dinner Party Problem. Consider a dinner party of 60 people and 6 tables of 10 each. How many possible seating arrangement are there? Or you can imagine 60 bricks making up a portion of an outdoor staircase. The point is we have 60 objects and 60 slots: this sort of thing happens all the time and doesn’t even encapsulate random continuous distances between objects.
Our host and hostess have a lot of possibilities to deal with. More possibilities than… the number of atoms in… the universe. That would be about 4x10^79 hydrogen atoms, counting only stars, which is most of everything. Our master of ceremonies has to consider a number about 200 times more than that:
Sixty people can seat themselves in 60! different arrangements across 60 seats.
60! = 8.320987113e81 = 8,320,987,112,741,390,144,276,341,183,223,364,380,754,172,606,361,245,952,449,277,696,409,600,000,000,000,000, which is a pretty big number.
I can imagine a mind boggling number of rocky planets located about 9 million miles from a somewhat larger than average yellow star. But I’m guessing that our infinite universe is a smaller infinity than that which would produce bizarro Earths populated mostly by Marilyn Monroe.
I concede that I have never studied different sized infinities or Cantor. So take my view with a hefty pound of salt.
Yet that number approaches zero when compared with infinity.
It has been observed many times that large finite numbers seem larger to humans than actually infinite numbers, even though they are, by definition, smaller. While this might tell us something interesting about the functioning of the human mind, it does not tell us anything about mathematics.
Exactly. On a planet with 3.5 billion women where I am the only man, my chances of getting laid seem infinite. My personality reduces that probability to a depressingly small finite number. If any of them see my penis in its flaccid state that number becomes zero and laughter ensues.
Right.
Even 8,320,987,112,741,390,144,276,341,183,223,364,380,754,172,606,361,245,952,449,277,696,409,600,000,00 0,000,000 factorial is tiny compared to even the smallest infinity (“countably infinite” or Aleph-null, the number of whole numbers, which is also the number of rational numbers. Anything you can put on a one-to-one-and-onto mapping with the whole numbers is this size. The number of points on a line is infinitely bigger. As I mentioned above, it gets deeper but I haven’t gone there.)
I don’t buy the “all Marilyns” because I don’t think it’s a possible outcome.
I’m not even sure I buy the “replica of Earth, just a little different” or even the same, for that matter, because I suspect that the complexity of the Earth might be infinite. But I’d have a hard time defending that proposition. I think it’s quite possible that I’m wrong, too.
Quantum mechanics doesn’t mean everything is digital.
Question for physicists: is energy quantized? Of course, a photon is a photon is a photon, but the energy in a photon is a function of its wavelength. Is its wavelength necessarily quantized? (It might be, based on the planck length.)
Even if distance is quantized, does that imply that direction is quantized? Could I move one planck length to the left, and then one at a 45 degree angle? I realize the resulting distance between the two couldn’t be expressed as a fraction of a planck length, but I doubt that’s sufficient to impose a “snap” grid on space.
Is it possible that any finite description of a (sufficiently complex) physical system isn’t sufficient to reproduce that system completely, including all subsequent behavior (ignoring quantum randomness, for argument’s sake)?
Oops, you are correct.
I thought I had some traction with the infinite numbers between zero and one analogy, but I grasp now that you are claiming that the boundaries are denoted by that which is possible, however unlikely. So while there are no planets where 2+2=7 (given certain axioms), a planet full of Groucho Marks impersonators is ok. I guess that implies that infinite rocky planets in habitable zones follows from infinite stars.
Q: What’s the evidence for a universe with infinite space? What’s the evidence for a universe with infinite numbers of stars? What’s the evidence for infinite multiverses as opposed to huge but finite numbers of multiverses? Is the evidence strong? I’m not being skeptical, btw: I simply don’t know.
It’s not really a matter of evidence for an infinite Universe. Rather, it’s just that there’s no evidence for it being finite, and the infinite models are simpler than the finite models. All we can actually say is that the Universe is larger than the portion of it we can observe; we can never actually rule out the possibility that it’s just slightly larger than that.
With multiple universes, it’s even more tenuous, given that we don’t even have any evidence of any other universes existing.
I think you are probably wrong. It is possible to conceive of a species where every female individual resembled Marylin Monroe to our eyes, yet were genetically diverse; just as all all chaffinches have the same pattern of colouration, yet are diverse. Perhaps if we were all members of the Hitler/Munroe species we could tell each other apart; but any other phenotype would look hideously deformed.
Here’s Max Tegmark’s reasoning about how far away the nearest identical observable universe might be.
http://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf
Hmm; he seems to have missed out a few possible characteristics there - what if all these subatomic particles were travelling in random directions at relativistic speeds? But I think this shows that the distance to a recognisably similar universe might be an estimable quantity, especially since the OP only specified millimeter resolution.
I prefer to think of it more as anything can happen; rather than everything will happen.