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Does Olber's paradox claim that in an infinite universe (both in space and in time) light should be able to reach you from any direction?

Consider the thought experiment of placing an infinite number of light-emmiting points on the interval [0, 1] on all the rational points.

In this case, the space is bounded, the number of stars is infinite but it would still be mostly dark (this is the well known mathematical fact that there many many many more irrational numbers than there are rational numbers)

Now if we transpose this idea to space.. I think we could easily have an unbounded space, homogeneous, with infinite number of stars but still it would be mostly black...

Sure, I can relax the condition that stars should be placed on rational grid points in the space.. And I also think one could get away with stars as balls and not points...

Or can't I?

Is there a mathematical proof of why this is a legitimate paradox?

It feels wrong to me.. But again.. My intuition is a random machine

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closed as off-topic by StephenG, GiorgioP, John Rennie, ZeroTheHero, Jon Custer Apr 22 at 14:53

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  • $\begingroup$ The paradox not only requires an infinite universe but also it should be both static and infinitely old. However, our universe does not seem to be static and is not infinitely old either, as we know it. $\endgroup$ – exp ikx Apr 18 at 14:42
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    $\begingroup$ I don't want to sound critical, but I'm having trouble understanding why you would think that it mattered whether the numbers describing positions were rational or irrational? Real numbers, sure. But what is the relevance of rational & irrational? $\endgroup$ – D. Halsey Apr 18 at 16:16
  • $\begingroup$ Because that was just an illustration for the well known mathematical “paradox” where you can paint infinitely many points with white on a black line and still the line is all black. And the way you do it mathematically is by painting all rational numbers (which are infinite) $\endgroup$ – gota Apr 20 at 22:33
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The original definition of the problem is the best: if space is infinite and there are an infinite number of stars, eventually every ray hits a star. This has the implicit assumption that stars are finite, so I think your thought experiment fails on that note - if there are an infinite number of finite spans on [0,1], as opposed to points, then everything is indeed covered.

Olber lived in a cartesian universe and the idea that you could be both infinite in space and finite in time simply isn't something he could have considered. But the universe is finite in time, and thus its not so much that the ray doesn't eventually land on a star, but that that star is physically not visible.

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  • $\begingroup$ +1 The point that stars are all of finite width is the important thing here. $\endgroup$ – Jahan Claes Apr 18 at 15:44
  • $\begingroup$ +1 but here is my crucial observation: we CAN put infinitely many balls (of finite width) in an infinite space and such that we have most rays never hitting a star! One simple (and obviously not realistic) way would be to put an infinite number of stars one behind the other on a single line. So it is possible to have infinite stars on an infinite space and have the sky mostly black. But if we assume homogeneity are we mathematically guaranteed that the sky will be mostly black? $\endgroup$ – gota Apr 20 at 22:40
  • $\begingroup$ @gota - indeed, but then, of course, you're violating noether's theorem (sort of, in reverse). $\endgroup$ – Maury Markowitz Apr 20 at 22:49

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