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Who needs infinity anyway?

Calvin Lin posted a query which was supposed to be about whether or not "completion of an infinite sequence" could have meaning.

If something takes forever...

but the subject ended up dwelling on the nature of infinity. Now I'm asking why do we need infinity at all? Here's the question being posed:

Imagine that we have two universes, one that necessarily involves the concept of infinity, the other not needing any such notion of infinity at all, but relying instead on "very large numbers". Is there any way we could tell the difference between the two? What experiment could we conduct to show that the universe that we live in is in one or the other?

Note: The concept of infinity doesn't have to be restricted to time or size, i.e., "infinitely old universe", "infinitely large universe". There are many other things that can involve infinities, such as, for example, quantum field theories.

Note by Michael Mendrin
2 years ago

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The hard part of this question is finding a physical phenomenal that requires an infinite amount of something (\(n\)) to have the result \(A\) as compared just a very large number of (\(n\)) which would give the result \(B\). And that (\(n\)) must be measurable, so it cannot be, for example, time or space (or can it?)

My guess is that we can't tell because there is nothing (or is there?) such that we can measure it as truly infinite rather than just a very large number Julian Poon · 2 years ago

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@Julian Poon One example would be black holes, which, in theory, has a singularity of infinite density. But do we really need that to have real black holes? Many physicists don't think so, especially when considering other factors such as quantum physics, which makes it moot. Michael Mendrin · 2 years ago

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@Michael Mendrin Wait, do black holes necessarily have a singularity of infinite density? Can't we have, say, just a body with such a high (bit finite) density that light (which has a finite speed) can't escape it? Daniel Liu · 2 years ago

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@Daniel Liu If we solely go by general relativity, it leads to the conclusion that black holes can exist, that have singularities of infinite spacetime curvature. Einstein was aware of this, and initially rejected the idea of black holes, even though his own theory predicted their existence. Einstein also initially rejected quantum physics, but at the end, it is quantum physics that makes possible a plausible model of black holes that avoids such infinite spacetime curvature. Stephen Hawking is famous for his work on this subject, along with Roger Penrose. Michael Mendrin · 2 years ago

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@Michael Mendrin What I mean by my statement was "if black holes necessarily need to have infinite space-time curvature at their singularities", not "if it is possible to have a black hole with infinite space-time curvature". Daniel Liu · 2 years ago

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@Daniel Liu Liu, first we need to distinguish between mathematical models and physical reality. Assuming we're just referring to the former, then we have to ask ourselves which mathematical model are we using? The concept of black holes originated in General Relativity, which does say that once a mass reaches a critical size and density (a variation of the Chandrasekhar limit) , it will gravitationally collapse and overcome nuclear forces, becoming denser than even a neutron star, after which no known ordinary laws of physics can prevent it from collapsing into a singularity of infinite spacetime curvature. Numerous physicists have had argued that certain processes prevented further collapse into a singularity, which is the reason why we have white dwarf stars, neutron stars, and even hypothetical quark stars. Regardless, given sufficient mass, once gravitational collapse begins, progress is inexorable into a singularity.

That is, if you strictly go by general relativity as worked out by Einstein. If you are asking if it's possible, per general relativity, to have a black hole without a singularity with infinite spacetime curvature, the answer is basically no.

However, alternative models, using variations of Einstein's general relativity or extensions of it using quantum physics, for example, of black holes have proposed that such a singularity need not exist. These models need observational verification, like other models in physics such as string or supersymmetry theory in particle physics. Sure, it's been possible to devise a mathematical model that avoids such singularities.

Here's such a paper that attempts to do just that, without straying far from general relativity

Black Holes Without Spacelike Singularities Michael Mendrin · 2 years ago

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@Michael Mendrin If it is indeed impossible for us to have an experimental setup that measures infinity, or to give us a result that only happens in infinity, then it should be enough to conclude that we live in the universe where very large numbers are sufficient. Julian Poon · 2 years ago

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@Julian Poon Proving a negative is always a lot harder than proving a positive. Just because we haven't yet found an instance where "infinity" is a necessary doesn't mean it never is. Maybe this is like Calvin's question about something taking forever to do?

Anyway, we don't necessarily need to try to measure "infinity" directly. There are some models in physics where infinities seem unavoidable, like quantum field theories. Feynman, Schwinger, and Tomonago jointly won the Nobel Prize in physics in 1965 for their work in quantum electrodynamics which entailed finding a way to deal with infinities that threatened to make any computation of probability amplitudes impossible. As Feynman himself explains frankly, they went with a process called "mass renormalization" which involves making an arbitrary "cut-off", akin to leaving out a infinity of decreasing terms in an divergent infinite sum. He was said to be deeply unsatisfied with that process, which he considered to be a desperate move--and yet, such computations yielded some of the most accurate ever verified by physics. So, are infinities needed, or maybe it is just an illusion? Michael Mendrin · 2 years ago

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@Michael Mendrin Wow. That's interesting, I never knew of these. Julian Poon · 2 years ago

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That's the problem of infinity. I firmly believe it's just human exaggeration if it exists then we should bother only when its needed like in black holes and the universe. I read in The Universe in a Nutshell that many infinities cancel out each other like the positive energy needed for creation of galaxies cancel with the negative gravitational potential. However more complex theories include more infinities about which I am not sure. If infinity were not there we would have crossed the speed of light Vishal Ch · 2 years ago

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I was thinking of time. We consider "TIME" as a physical quantity. Sir, don't you think time can be infinite. Although I must say that my way of seeing time is not very much developed. Rishabh Tripathi · 2 years ago

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@Rishabh Tripathi Not only your way of seeing time may not very much be developed, physicists still haven't quite figured out what time either is or why we have it, and many believe it to be an emergent property. Indeed, some theories about black holes say that time could be conflated with space. In that case, if we cannot even measure when time starts and end, we probably can't even speak of time as being finite or infinite. That may ultimately not have much meaning, to try to make that distinction. Michael Mendrin · 2 years ago

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@Michael Mendrin Well... conflating space and time is a theory. It hasn't been experimentally proven ( as far as I know). The time we're living in, is this real ? Isn't it measureable and if we take the big bang as the starting point of time, then it must be having some value in some units till now and that value would be increasing continuously. By saying this, I'm actually visualizing time as an endless string, one end of which is at a fixed point but the other seems to be at infinity. Rishabh Tripathi · 2 years ago

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@Rishabh Tripathi An early "conflator" of time and space was Albert Einstein, remember? He was one of the first to suggest that time and space weren't inherently different. Michael Mendrin · 2 years ago

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@Michael Mendrin But sir, isn't it true that he was a very strong opposer of \(\text{Quantum Mechanics}\)? Sravanth Chebrolu · 2 years ago

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@Sravanth Chebrolu One of the great ironies in the history of physics is that the one time Einstein won the Nobel Prize in Physics was for his work on the photoelectric effect, which demonstrated the quantization of light energy, which paved the way for the development of quantum mechanics. Yet, he lived to hate what his experiment had spawned. His battles with Niels Bohr about the "absurdities" of quantum mechanics lasted for decades in the early 20th century (even though they were otherwise good friends). Most famously, Einstein proposed to Bohr a thought experiment intended to bebunk quantum mechanics as an impossibility, the EPR quantum non-locality experiment, which was actually later proven experimentally to be fact. Michael Mendrin · 2 years ago

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@Michael Mendrin Yeah. It might have seemed illogical for his brain. It not a pleasant surprise that Einstein didn't accept the theories of quantum mechanics. . . Sravanth Chebrolu · 2 years ago

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@Michael Mendrin But then why we still study Quantum Mechanics ? It can't be totally absurd. Well.. it may have many controversies and uncertainties. Rishabh Tripathi · 2 years ago

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@Rishabh Tripathi Many claims made in quantum mechanics sounds preposterous, but it's produced some of the most accurate predictions in the history of science. So, we just have to accept it as a experimentally confirmed fact, and try to find a way to have it make sense to us. Michael Mendrin · 2 years ago

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@Michael Mendrin Yeah.. his theory of General Relativity. It was about how the space-time acts like a fabric. Its deformity causes the gravity and other phenomena. But the question is: If we leave the black holes apart (because we really don't know much about them), can't the space-time fabric be measure in some way ? ( well I don't know much about it and just asked in curiosity. I am not proving the point that infinity exists.) Rishabh Tripathi · 2 years ago

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@Rishabh Tripathi If you're asking if curved spacetime can be "measured in some way", hell, all GPS now have to carefully take into consideration how spacetime is warped around the Earth, or else GPS readings will be all off. Accurate GPS depends on super-precise measurement of time intervals, and local curved spacetime messes it up. It is already a practical reality, staring right in the face of engineers trying to develop such systems. Michael Mendrin · 2 years ago

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@Rishabh Tripathi Time has not been defined and probably can't be defined because of its odd nature. Even if it were infinite we should not care much about before the big bang Vishal Ch · 2 years ago

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Well... I think its just our lack of knowledge that leads us to the concept of infinity. Mathematics is yet to develop to define it. Its same as when the concept of Quantum mechanics was coined. When people didn't understand something they called it quantum. But now it has been defined and rules have been set, though we're still making progress in it. Rishabh Tripathi · 2 years ago

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Yeah. I agree with Julian sir's argument. We might expect time to be never ending. But space may be considered to end at some point, but we do not know whether it ends. It's a hypothetical question. If someone asks, "what's the biggest number you know?" to a grade \(1\) kid, he may reply a \(100\) or \(1000\). But this doesn't mean there are no numbers exceeding \(\text{100 or 1000}\).

But, if the same question was asked to you, what would you reply? Some may say I honestly do not know. While others may argue, if given infinite time I'll tell you infinitely many numbers(as in the case of Calvin sir's post). A better question, might be to ask, what exactly is meant by \(\text{"relying on very large numbers"}\). Again here, it may depend on the person as to what exactly he considers \(\text{"very large numbers"}\) to be.

We may consider the case of never ending decimals such as the one post by Jake Lai, which proves that: \[0.999999 . . . . . . \infty = 1\]

This may show that the thing the other universe consider as \(\text{"very large numbers"}\) is actually nothing but infinity(or is it?). Sravanth Chebrolu · 2 years ago

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