substitutematerials

So if it is infinite and uniformly filled, but not repeating (c), there is definitely a moon with invisible unicorns right?

To push this question back towards cosmology, can we ask what would fill a infinite universe? The possibilities I can think of are: a.) we live within a finite section of an infinite volume which contains matter, which is surrounded by infinite emptiness. b.) We live within one of a finite number of such finite sections in an infinite volume. c.) the universe is homogenous and isotropic on an infinite scale, and all configurations of matter exist in it, including a bizarro Earth where invisible unicorns live on the moon d.) the universe is homogenous and isotropic on an infinite scale, filled with an infinitely repeating pattern of precisely the same configurations of matter we observe in our observable universe Is this list of possibilities exhaustive?

Alright, I'll take it. And your point that scientific theories are rarely abandoned completely is valid. I can think of a host of abandoned theories that are prescientific or were only ever quasi-scientific, but it depends in part on where you draw the line for Science right? We've tossed out the four humors in medicine, geocentrism in astronomy, the four elements and alchemy in chemistry, phrenology in biology, homeopathy in medicine... But like you said, most of these were probably abandoned more due to contradictory evidence than internal contradiction. Can we close by returning to this narrower question: In the case of a finite universe (bounded or unbounded), could we define the aggregate rate of spatial expansion by [math]d*H_o[/math] Imagine that we obtain a really fantastic measurement of curvature, and the universe is curved right at the threshold of what we can't observe right now- so implying a diameter of 14 trillion light years or 4292419 megaparsecs as per the article mentioned. Why couldn't I multiply that diameter by the present rate of spatial expansion, 72 km/s/megaparsecs, and state that the diameter of the universe is increasing by 309,054,168 km/s? This rate might have no use whatsoever, but wouldn't it be logically implied, when we assume a scaling factor and finite diameter?

I can virtually guarantee you that there is a scientific theory, accepted presently, that will be shown to self-contradictory at some future point. If I knew which theory this was, I would not be lurking in science forums. Furthermore, mathematics and logic are the foundations of the physical sciences, so the things you can say about them necessarily also apply to science, although not necessarily the other way around. My example of Euclidean mathematics discovering internal contradictions most definitely has relevance to scientific models. A historical example from the physical sciences of self-contradiction could be Phlogiston theory, which ended up suggesting that phlogiston had both positive and negative mass- an internal contradiction that helped undo the theory.

Why you gotta be like this Strange? If your time is being wasted, you are wasting it, not me. Your diligence in responding to these forums is appreciated, but you are not required to. And on these points you are wrong. Thompson's lamp relates to the fact that there is a maximum speed that operations can occur in reality, as dictated by special relativity. And the painter's paradox illustrates that real objects made of matter are discrete and quantized. And come on dude, the quantization of light to resolve the ultraviolet catastrophe doesn't relate to science? Your assurance that all scientific theories are obviously self-consistent is contradicted by Godel's second incompleteness theorem at least, isn't it? I agree MigL, A beginning to time would be an edge. And the combination of the big bang and the second law of thermodynamics sure make it look like time has a beginning.

Thanks Outrider, The jury is definitely out on this very old question, I'm just sharing the things that have shaped my intuition. I think your 2 additional concepts are equally vexing- something from nothing, or why did the universe start, might be truly unanswerable. As for what lies on the other side of the universe, there are several ways of looking at this: Firstly, there can be a boundary condition from any observer's perspective, which is easy to achieve in an expanding universe. No observer can get any closer to the receding edge by real travel, so there is no way to interact with such an edge. Like the event horizon of a black hole from the inside, a causal boundary. The more popular solution which we've been discussing here is boundlessness, where travel in any direction will eventually return the observer to the place they started, like traveling on the surface of the Earth. And my least favorite option which just kicks the can down the road, the boundary of our universe is other multiverses or a parent universe. That's not really fair Strange. I presented paradoxes as asked, as well as an important example of infinity disappearing from science history. Of course I'm not going to prove here that the universe is finite, I'm just telling you what informs my intuition. It is most definitely possible that the universe is unlimited, I'm just leaning in the other direction. I read about this self-consistency principle a bit- it's interesting, although I'm not sure I understand the mechanism through which it is proposed to operate. I would say that our world is pretty fantastically chaotic, and that the idea of a whole live human being going back in time and not generating any paradox-inducing situation at all is a challenge. The grandfather paradox usually talks about killing your grandfather, but heck, all you'd have to do is add a one second delay to the day he and grandma conceived your parent, and you'd probably still prevent your existence because a different sperm would make it over the finish line. And culture is just as stochastic. The past is so elaborately determined that there would not be a lot of room to move around if something was preventing any paradoxical situation from occurring.

Sure thing. Thomson's lamp: A lamp is turned on or off when half the time left until midnight has elapsed. If the lamp can be toggled instantaneously, the switch will be toggled infinitely many times as midnight approaches, since we can always divide the remaining time in half. What state does the lamp end up in at midnight? For the abstract instaneous lamp, there is no answer. Real lamps, however, have a minimum amount of time to flip the switch, which would render the answer simple. Painter's paradox: Take a mathematical object like Gabriel's horn, the rotation of y= 1/x around the x axis cut off at x=1. Calculus shows us that such an object encloses a finite volume but has an infinite surface area (isn't calculus neat). Can you paint the inside of Gabriel's horn with a finite amount of paint? On the one hand, obviously no, since the surface area is infinite. But if you completely fill the finite horn with paint, won't you also paint the entire surface? Again the resolution is that paint is a real physical thing that is not infinitely sub-dividable, it's made of molecules with a size, and at some point the horn will be too thin for them to pass and part of the surface will remain unpainted. In truth anything real thing that horn could be made of would have the same restriction. Gabriel's horn is an abstraction. The most important scientific removal of infinity that I know is the Ultraviolet Catastrophe. Unless we quantize the emission of light into photons, blackbody objects should emit an infinite amount of energy, which obviously they do not. Infinity is a limit, not a number. There is no known instance of it in nature, so I would be surprised to have this one notable exception regarding spatial extent.

Previous scientific systems such as pythagorean mathematics were undone by self-contradiction- the premise that "all phenomena in the universe can be reduced to whole numbers and their ratios" was shown to lead to paradoxes. We're also not being rigorous about what incomprehensible means in this thread and we should keep that word out of the conversation. Let's stick to discussing the self-contradictory or paradoxical, that's what I'm trying to talk about. And there is a pretty big list of paradoxes associated with the idea of infinity, suggesting it is not a physical concept but purely abstraction.

It leads to unresolvable paradoxes like the grandfather paradox. These paradoxes suggest that single timeline time travel isn't possible, because it leads to situations that don't make sense, i.e. are not comprehensible.

Single timeline time travel? The most fundamental requirement of a scientific theory is that it be logically self-consistent, right? Internal paradox renders an idea fundamentally incomprehensible.

Ha yes! Comprehensible by who, that's the rub. I also don't understand the maths of GR, but I trust that others out there do. This is different than something which everyone agrees is fundamentally incomprehensible.

You're right, we don't know that reality itself isn't infinitely subdividable. But everything we do know about it points to quantization, that's particle physics. We know that if a spectrum of blackbody radiation were infinitely sub-dividable, it would contain infinite energy as per the ultraviolet catastrophe. Zeno's paradox also argues against infinite subdivision. I think you are allowing a broad fallacy of 20th century science in your second point. I think it stymies critical thinking and the pursuit of new models if we accept an incomprehensible reality. "Making sense" is an important part of our tool box. If a scientific model isn't comprehensible, I assume it is provisional, no matter how well it generates results. "Shut up and calculate", as quantum physicists sometimes say, is a bad attitude.

Many laypeople have trouble imagining a finite universe; I'm a layperson who struggles more with an infinite one. It seems to me that infinity is purely a useful mathematical abstraction. Just as reality is not infinitely subdividable, i.e. it is quantized, I can't imagine it being infinite in extent. Especially if we adhere to the Copernican principle and assume we are in a representative section of the universe, an infinite universe seems absurd to comprehend. We couldn't do calculus without the abstraction of infinite subdivision, but there is math that requires the square root of -1 as well. These abstractions are useful but non-real.

Those are both good points Strange, especially the uncertainty about the physical existence of singularities.

Ah this is good guys thanks. I had thought that perfect flatness necessitated an infinite space, without realizing the the cylinder and 2 torus are geometries that are flat and finite. Imagine that we obtain a really fantastic measurement of curvature, and it is curved right at the threshold of what we can't observe right now- so implying a diameter of 14 trillion light years or 4292419 megaparsecs as per the article mentioned. Why couldn't I multiply that diameter by the present rate of spatial expansion, 72 km/s/megaparsecs, and state that the diameter of the universe is increasing by 309,054,168 km/s? I have to let some crankiness rip here though: the precise observations of apparent flatness make FLRW space look endangered to me. Sure, flatness is a possibility in FLRW space, but we have no explanation for how mass-energy should be so precisely balanced to achieve this. Include the acceleration of expansion, and you've got a universe made of 95% undetectable mystery sauce balanced on the head of a pin. I'm also curious what you think of my suggestion in post #9, that a singularity is a spatial edge, akin to a needle punching a hole in a sheet.