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Obviously, one is a sphere and one isn't, but would the inhabitants of either world be able to rule out the possibility that their world might actually be the other?

We humans knew that our world was spherical because of its curvature, but the larger the sphere the less the curvature of its surface so it stands to reason that in an infinitely large sphere there should be no curvature (or rather it is infinitely small and indistinguishable from none).

Since they can't coexist with infinitely large worlds, assume that either there are no celestial bodies (sun, stars, other planets etc.) or that they depend on the location of the observer (there will always be a sun visible exactly above one's head, or perhaps the world has a luminous "ceiling" above the atmosphere)

Assume no space travel but arbitrarily precise measuring instruments and arbitrarily high resources dedicated towards finding out whether their world is a sphere or a plane.

Assume earth-like conditions (gravity etc.), on the sphere gravity will pull towards the center, on the plane it will pull down.

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closed as primarily opinion-based by Mołot, Aify, Mormacil, John Dallman, Hohmannfan Jun 25 '17 at 20:15

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ As neither is real, difference is only philosophical. $\endgroup$ – Mołot Jun 25 '17 at 8:06
  • $\begingroup$ Is there movement of the Sun at all? Or changes in the 'luminous' ceiling'? $\endgroup$ – adonies Jun 25 '17 at 8:06
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    $\begingroup$ Can you define "infinitely large spherical world"? I'm having a lot of trouble envisioning this. $\endgroup$ – Aify Jun 25 '17 at 8:10
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    $\begingroup$ How can you have a sphere if the radius is infinite? You wouldn't have a surface. By definition, a sphere has to have a surface, which means that there is a center, and therefore a fixed radius. $\endgroup$ – Aify Jun 25 '17 at 8:14
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    $\begingroup$ No, it's not "hypothetical". It can't exist, not physically, not theoretically. Infinity is not really a number, it's only something you can realistically use with things like $lim$ $\endgroup$ – Mołot Jun 25 '17 at 8:55
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If rocks become harder as you dig deeper, you are (probably) on a flat world.

My first thought was "dig". Obviously, if you can dig through the "bottom" of the planet then you're living on a flat world. But you don't even have to go that far.

Neither world makes sense according to physics, so we should presume this world is a simulation. Therefore, we must take into account the rules of the simulation. Either the simulation takes pressure into account, or it does not.

If the world is an infinitely large (or even merely "very" large) sphere and pressure exists, the pressure should increase as you go further down, becoming magma before finally turning into exploding plasma above an infinitely large neutronium layer that eventually builds up into an infinitely large singularity. Now, since this concept makes no sense it is unclear what effect it would have on the world, but what is clear is that since this world is effectively a thin layer of rock sitting on top of an infinitely large exploding star it is probably not going to be habitable.

Since the world is habitable, it stands to reason that, if the rock is infinitely deep, pressure is not being simulated. That means that the ground should not become harder as you go down, but remain a uniform, infinitely large rock-filled space throughout. This also means no volcanic activity, which means no metamorphic (or igneous) rocks.

A flat world, on the other hand, can have normal pressure without becoming uninhabitable, since there is a limit to how high the pressure becomes, and this limit is probably above the point where fusion is initiated and the planet becomes a star. Therefore, pressure can increase as you go down, allowing the flat world to have normal, Earth-like geography up until you reach the bottom. Below this point - the "bedrock", so to speak - there is effectively nothing. Therefore, if pressure increases as you go down, - indicated by the presence of igneous or metamorphic rocks - you are living on a flat world.

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    $\begingroup$ "If pressure increases as you go down, - indicated by the presence of igneous or metamorphic rocks - you are living on a flat world." but if it doesn't increase, it may again just be that pressure is not simulated, as in the spherical case. Further, in the spherical case, since it is a simulation (by your logic), the simulation of pressure may reach a limit at some point well before (or just before) neutronium or a singularity: Once pressure and heat are sufficient to ensure further digging is impossible, the simulation could stop there! Thus digging tells us nothing either way. $\endgroup$ – Amadeus Jun 25 '17 at 13:23
  • $\begingroup$ @Amadeus True, it isn't proof, but it is the assumption one would make using Occam's Razor and the Cosmological Principle. A pressure limit above that of a singularity at an arbitrary point within an infinitely large sphere suggests an additional law of physics we have no reason to believe exists, and therefore we should presume it does not unless we have evidence to the contrary. $\endgroup$ – IndigoFenix Jun 26 '17 at 7:48
  • $\begingroup$ You weren't talking about laws of physics, you were talking about a simulation; in which we can make the laws as arbitrary as we want. You posit the simulators must and did violate the laws of physics to be on a sphere: They could as easily violate the laws of physics on a flat world. You can't posit they must do it on one, and must not do it on the other! That is illogical; The densities they create on the sphere could just as easily be duplicated on the flat world; and then also be equally infinitely deep. $\endgroup$ – Amadeus Jun 26 '17 at 11:32
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Is there a difference?

Not by any practical measure.

A planet with infinite curvature will curve if you manage to go infinitely far: it bends around a point infinitely far away, which is someplace! A flat surface, however, will stay flat, no matter how infinitely far you go.

Infinity is a very big idea, though, and we note that us mortals (and certainly the scientific and engineering mortals) do not deal with infinities anywhere except in our minds and through mathematical constructs. Sometimes we call things "practically infinite" to make life easier, but infinities is still something we do not deal in. Anyone living on this infinite-something-shape place, assuming similar physics to our own, would not be able to tell the difference.

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    $\begingroup$ As a side note, and therefore a comment: there are some interesting things that could happen in this world, like constant gravity no matter how high up you are (much like the E-field of an infinitely large plate stays constant no matter how far the test charge is from it). Neat to think about, terrible to actually live in! $\endgroup$ – PipperChip Jun 25 '17 at 8:22
  • $\begingroup$ Can you be sure: "terrible to actually live in!" Why it might be infinitely nice. I like the constant gravity idea. It would really ruin space travel. Would the curvature of space, as in General Relativity, be reflected in spacetime around either the infinite plane or the infinite sphere? The notion that mass might be infinite too in either case is a worry. $\endgroup$ – a4android Jun 25 '17 at 8:30
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NO.

They cannot tell. No matter how sensitive their instruments are, there is a limit to the sensitivity of any instrument, and by claiming the world is infinite, any change in curvature (or direction of gravitational pull) is infinitely small, and beneath the measurement sensitivity of any finite device.

YES.

Since we have already introduced in this story magical infinities; we might as well say the finite instrument is infinitely sensitive, so of course in a matter of a few yards it can detect the infinitely slight change of angle in the directional vector of gravitational pull, and with a series of such measurements conclude it is on an infinite sphere.

Who is Measuring, Again?

In the setup of this question, other planets and stars cannot coexist with this planet because it is infinite in extent. Well neither can space, air, or inhabitants, for the same reasoning! If I stand on a planet, my eyes are six feet above its surface. How is my head, held six feet above the surface of the planet, any different than a moon or another planet? It is not. By the rules in the setup, if the size of the planet prohibits anything else from being in space due to its "infinity" then it prohibits air, inhabitants, or anything at all from existing upon it. By the rules of mathematical induction it cannot even have a surface because such a boundary divides one space from another (say inside the sphere from outside the sphere) which violates the premise that it is infinite.

Which means...

Of the two options given, only the flat world can actually have inhabitants, air, sky. They cannot be on an infinite sphere and also exist, but they could be on an infinite plane and exist.

Alternatively, they could be on an infinite surface that happens to be flat in their observable region; but on some unimaginable (but finite) scale undulates or curves. If they are locally flat, then even infinitely precise instruments would not measure any curvature. (Understanding that "locally flat" could mean flat for millions of light years in any compass direction.) (Also, the same could not be true on a sphere, or it would no longer be a sphere.)

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