A note on the Universe's Bounds
By Tom Brown
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Recently on the radio the question was raised whether the Universe could be infinite. The discussion was on astronomy and astrophysics. The studio guest a respected professor, simply dismissed the idea as nonsense and he offered the following simplistic argument.
Argument
“When you go outside on a cloudless night, if there were infinitely many stars the whole sky would be white”.
The consensus is clearly that infinity theories are left to a lunatic fringe and simply do not deserve any serious attention. An easy way out. His “proof” was in effect based on the total observed brightness.
My intention was to write the essay in such a way that the text is selfsufficient, sound and consistent. The little bit of mathematics really is meant only as enrichment and entertainment.
Counter Argument
First Example
His argument is not valid and I will describe a hypothetical situation to refute his “proof”. For the construction suppose all the uncountable stars of such a universe were lined up as on a ruler on the familiar number line, stretching endlessly from opposite sides of our star.

A scientific measure for “apparent” brightness is additive and it obeys the familiar law of being directly proportional to the inverse of distance squared, where the distance is from the light source.
L ∝ 1 / R² ( brightness ∝ 1 / distance² )
By additive I mean that two lights together give their sum of their measure of brightness.
Suppose that from the Earth we would be in sight of every star, and each one as well as our neighbouring ones are identical and have the same luminosity.

To calculate the total apparent brightness L_{total} my reasoning is as follows:
You only need to look at one direction, the other would be identical because of the symmetry. Let the apparent brightness of the neighbouring star be L. Because they are evenly spaced the inverse square law says the next star's brightness must be 1/2² L or 1/4 L, since it is twice as far. The third one is three times the distance so it's 1/3² L = 1/9L and the next 1/4² L = 1/16 L etc.
So that the combined brightness of the uncountably many stars,
L_{total} = L + 1/2² L + 1/3² L + 1/4² L + … + 1/n² L + …
= L + 1/4 L + 1/9 L + 1/16 L + … + 1/n² L + …
= L (1 + 1/4 + 1/9 + … + 1/n² + … )
The sum of the infinite series in the brackets is a finite number, the series converges. This proves that the total apparent brightness is not infinity. In fact it even is less than 2L.

This is then one possible pattern. There are many many others, from the simple to very complex. Even should one postulate just a kind of regular ( in some or other way uniform) random distribution of stars you would most probably find the same and the analysis is not much harder.
There are many such conceivable configurations that would easily give the same result. If you really want to take it to the limit you could for instance use our known part of the Universe as just a building block instead. It wouldn't really matter in my construction since after all infinity is very large and very patient.
Interestingly in my example we have a static situation. All is in balance because of symmetry. Each star experiences exactly the same (distinct) gravitational attraction forces, the sum of the forces either side is finite because gravity also obeys the inverse square law, and the resultant (combined) force on every star is zero. The reasoning is similar to that for light intensity.
There is a brief discussion on infinite series (sums) later on.
Clarity on the Ideas
Actually there are a few distinct scenarios that could be seen as in one way or the other “boundless” or “endless” or “infinite”.
To start with we have to qualify what is meant by the Universe. One definition could be the sum total of all matter energy and space and time. I don't think it is really possible to define but we could work with that.
Let us examine and clarify three distinct possibilities. In each we think of the universe as static, frozen at one instant in time, a snapshot. Of course this is not the nature of time as we know from modern physics, but it works for me.

The most basic idea is the universe that although all matter and energy might be confined, space itself is not restricted. This sounds obvious but I've read some magazine articles which prohibits even such unbounded space on grounds of considerations of general relativity and the “bigbang”.
Even though it is materially bounded still there would be endless empty space, as on the geometric number line or plane or threedimensional Euclidean space. This idea would be the most widely acceptable and it is closest to our every day experience. Intuitively it makes sense, most of us think of space in this way.

Another type of endless universe could be to still have only a finite amount of mass/energy, but spread out in space in a way such that the material cannot be confined. That is there are no spatial constrictions on the distribution of energy and matter.
Although it is feasible as a mathematical possibility this is not really conceivable for the reason that mass consist of particles and perhaps there is a particle of least mass, a “smallest” particle. In that case it wouldn't be possible to spread a finite mass over an endless space. It is easy to prove because then there would have to be a finite number of particles.
I do believe it should be that there is such a least, smallest particle, so that intrinsically, simple mathematics would have to prohibit this idea. Although apparently neutrinos have no mass or energy. (Do they qualify as matter?)

One may also think of an infinite mass, but contained in a restricted space. This doesn't sound very likely even only on geometrical grounds.
 
Then finally, and what I would like to defend, is the idea of an infinite mass distributed more or less equally but continuing endlessly on and forever through space and time. In effect Uncountably many stars.
Some other Models
If I may indulge in another Einstein “thought experiment”
Second Example
As a more concrete example let us consider the following hypothetical situation: A very simple model with two stars moving away from the origin in opposite directions each at very high velocity. This would represent an expanding universe. If the relative speed is higher than the escape velocity (which would depend on the respective masses) there will be no coming back, no contraction or falling in, and these bodies will simply keep on indefinitely separating further and further.
The point is that whatever spatial bound you place, in time it will be exceeded and it has to mean that although at any instant in time this model universe is bounded, seen as a whole it is not. So that without taking into account limitations of time it is in fact in space, unbounded.
The idea is very familiar from high school algebra one might think of some graphs parabola, hyperbola a straight line and so on.
One could also think in terms of the familiar simple idea: However far you travel into space no matter how far ever it should be, it must be possible to go even further, and (empty) space simply has to just carry on for ever.
I am aware that in some models based on General Relativity space is kind of closed and folded into itself (convoluted) so that this need not be true, though to me it's hard to believe.
Third Example
Let us visualise a universe consisting purely of logic and mathematics. The question is whether this universe is infinite, or if there are essentially only so many basic truths. Is the sum total of all provable theorems bounded?
For instance in the proof of some central theorem do all the foregoing results, lemmas and propositions, constitute a part of the proof, or are they distinct truths?
A proof by the principle of mathematical induction is already in itself the culmination of infinitely many very small proofs yet in the end there is only one generalised result.
So the question is what does constitute a distinct result? The problem that one has is repetition. For instance it would be hard to know whether there could be one ultimate generalised theorem of which all others are simply special cases. People working in in the field of logic and set theory could probably give one a better understanding.
Another even more fundamental problem also related to repetition is the question of abstraction, and limitations it imposes.
Could you go as far as to reason that since every abstract mathematical structure is developed entirely on a handful of basic assumptions (axioms), do those few axioms in themselves already constitute all the results and the truths that can be deduced? In this case the whole show collapses into triviality.
I believe the answer lies in chaos and complexity and in studies of probability theory and even the very simple basic ideas. And to me that answer must be Yes. The universe of thought and of logic is in fact boundless and endless.
Cosmic Background Radiation
The observed (measured, and charted) microwave radiation background is believed to be crucial evidence for an historic event called the “BigBang”. In fact CBR has even been cited “the holy grail” of cosmology. I want to speculate that an endless (material) universe could explain this cosmic background radiation too. My knowledge on this is very limited and for the present I would certainly not venture into such territory.
However I believe there are serious anomalies in modern cosmology, for instance comparisons of the measurements of redshift in faroff galaxies and their respective distances. To me it looks as if we are reasoning in circles.
Further, I don't believe that in considerations of inflation the fact that the (local then!) Universe is expanding should really give rise to any serious difficulties. Certainly in the mathematics and geometry it could be easily accommodated.
Hidden Assumptions
Has anyone ever seriously attempted to show it cannot be so, that the material Universe must of necessity be bounded? Has anyone tried to prove the Universe is finite? The point is simply noone would bother, we all know it is and it is an accepted truth and that's it and we may freely scoff at any such absurdities.
Clearly we are dealing with a blatant hidden assumption.
To illustrate the idea of this type of reasoning tautology. If a person wishes to prove that there can be no miracles his starting point simply has to be precisely that miracles are not possible. It has to be so, and of course as such an argument as a whole then is null and void. Because very obviously, if a miracle can happen then it means anything can happen. No?
There are no logical or practical grounds to believe “supernatural” miracles are not possible save only for our own petty experience which honestly is hopelessly inadequate and totally insignificant. I cannot see scientific or logical grounds to deny the possibility. It boils down to a question of faith from either side. It is therefore as much rational to believe in miracles, as not to.
“There is a principle which is a bar against all information, which is proof against all arguments and which cannot fail to keep a man in everlasting ignorance that principle is contempt prior to investigation.”
—Herbert Spencer
Conclusion
In these hypotheses I have as yet no scientific explanation or theory of the origin or start of material creation. The nearest I can come is to say that perhaps it always has been and always will be.
I do maintain strongly that there is no more reason to believe that the Universe is finite, than to believe it is infinite. The question really belongs more in the realm of philosophy than of science.
∞
∞  ∞  ∞
What follows are brief discussions on some relevant ideas.
On the Examples
Unbounded functions
Related to the Second Example, a function (graph) can be unbounded even though at every point the value y is a real number not infinity. To explain the idea think of the behaviour of y = ½ x. It is a straight line through the origin with constant gradient (slope) of m = ½. The angle with the xaxis is 26.6°. A more interesting graph, it steadily becomes steeper and steeper, is the parabola y = x² + 1.
In both functions every number substituted for x results in a (finite) number y.
But let us try a number, say y = bound = 10 000. Now pick a large enough number, say x = 21 000. The yvalue, height, on the line at 21 000 is 10 500 and it surpasses our candidate bound of 10 000.
Let us try a much larger number as a bound for the parabola, y = bound = 4 000 000. On choosing x = 2013 the height is y = 2013² + 1 = 4 052 170 and it is larger than our candidate.
If you draw the graphs you can see it. As a whole either graph will eventually pass through any horizontal maximum that you might conjure up.
Since we are actually interested in time rename the xaxis t for time, x = t, and think of y as distance, y = ½ t ; y = t² + 1. The distance cannot be be contained, thinking of time as the whole of the taxis (eternity). For any conceivable restriction on height or yvalue if you wait long enough it will be exceeded. There can be no maximum and these graphs are unbounded.
A graph can even be unbounded in a finite time. Consider y = 1 / ( t – 2 ). This is a hyperbola with a horizontal asymptote at y = 0 and a vertical asymptote at t = 2. The yvalue as well as the slope increases rapidly as t approaches 2 from the left, and then instantaneously explodes to infinity.
All these graphs illustrate a bounded but eternally expanding situation.
Infinite Series
The familiar geometric series 1 + r + r ² + r ³ + … is convergent when 0 r
∑ ½ⁿ = 1 + ½ + ¼ + ⅛ + … = 2
This type of sum occurs frequently in everyday life. Unfortunately there usually it is not bounded this depending then on whether you are a bank or a borrower! Think in terms of compound interest.

In the First Example the crucial question is whether the sum of squares
∑ 1/n² = 1 + 1/4 + 1/9 + 1/16 + …
adds up to a finite number. To find the exact total one may add this infinite series which you can, since it is convergent and must be a finite number.
 
There are advanced tools such as Laplace transforms and residue methods in Complex Analysis to evaluate infinite series such as these.
Please note it is not necessary to actually calculate the value in order to know whether the series has a finite sum. There are simple tests in a standard firstyear syllabus which can determine most types of series as convergent or not, or ± ∞.
Reality and Mathematics
Mathematical Induction
Induction is a method of proving certain theorems. The idea is a bit sophisticated but in essence it is simple.
The easiest way to visualize it is a very long row of dominoes (tiles) standing upright one after another. If a domino tips and falls it knocks down the next one too, and that knocks down the next. Then should I knock over the first one there is an endless chainreaction and all of them fall, one after the other.
As an application I propose to briefly prove a formula for an arithmetical progression. It is a standard exercise.
Prove the formula : S (n ) = 1 + 2 + 3 … + n = ½n (n + 1)
First assume that this formula holds for n = k, i.e. assume the equation is true for some integer k. By using this, if I take the next number n = k +1, I must show that
S (k+1) = 1 + 2 + 3 … + k + (k+1) = ½ (k+1) ( (k+1) + 1 ).
LHS = S (k+1) = [ S (k) ] + (k+1) = [ 1 + 2 + 3 + … + k ] + (k+1)
= [ ½ k (k+1) ] + (k+1) = ½ k² + ½ k + k + 1
= ½ (k + 1) (k + 2) = RHS
The formula works for n = 1 because ½·1·(1+1) = 1.
By the principle of mathematical induction we may conclude that our formula works for every positive integer n. The first domino is the case n = 1, and if any domino is named k, the next one would be k+1.

This kind of application is commonplace we learn it already in secondary school. There are very advanced and sophisticated theorems that are proved with induction. It is an indispensable tool.
The point is that the proof of such a result in fact consists of infinitely many lesser theorems. That is, the truth for n = k implies the truth of n = (k+1). Is it then one result, or infinitely many? Even if you might say that as such it is only one theorem, still in my opinion the academic knowledge comprising the total of all possible provable truths is unlimited.
Chaos and Complexity
If someone might ask me, “Where did you get this story?” I may truthfully answer, “From the dictionary”.
Take the Oxford dictionary and completely at random pick one word, and write it down, and pick another at random, and write it down, and another random word and carry on with this process indefinitely. Somewhere along the line you would find “take care of the sense and the sounds will take care of themselves”. It is because there is a very very minute but nonzero probability that the words will appear and in this sequence, it will pop up sooner or later, or much later.
Following this reasoning at some stage the King James Bible would appear and Shakespeare's complete works too, but you would probably be at it for all orders of astronomical orders of time. To get even something like the poem “the little red wheelbarrow” could take centuries.
Of course it is not feasible. Someone might try to do a computer generation but that's just as futile. Apart from the machine's pathetic inadequacy someone would have to read all that junk otherwise you'd never know. Can you imagine how must junk must be in between?
All English literature past present and future all knowledge every encyclopaedia is contained in such an infinite sequence of words. It has to be infinity. This is just one sequence of words. To me it is enough proof that knowledge has to be literally endless, and with it, all provable truths.
Abstraction
One, two, three, four, five, six, seven, eight, nine, ten.
Eins, zwei, drei, vier, fünf, sechs, sieben, acht, neun, zehn.
1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10
Two of two of two is eight.
Zweimal zweimal zwei ist acht.
2 × 2 × 2 = 8
one, two, three, ? six
eins, zwei, drei, ? sechs
1, 2, 3, ? 6
The idea I wish to convey is the fact that the structure of arithmetic is inherent to the numbers and operations and is completely apart of how it may be described. A fundamental philosophy in modern mathematics is reducing systems to their core structure. It turns out that the method of abstraction is immensely powerful.
In deciding whether mathematics and logic have finite, uncountably many or even more ideas there is another serious difficulty the duplication of results and theorems.
“What's in a name? That which we call a rose
By any other word would smell as sweet.”
—Shakespeare
The nature of Time
One absolutely crucial question is whether the Universe had a beginning or not. Does time had a starting point? Was there a beginning of the material universe? Or has it always been, has the universe existed always? As Hubble himself realised, if time had a beginning or start it has to correspond to a material creation event, such as a bigbang if you then wish.

A semigroup can describe this causality, where time has a beginning and history cannot be retraced. Very simple philosophical and practical notions also are encapsulated in the essential properties that follow.
E (0 ) = I ; E (s + t ) = E (s ) E (t )
E(t ) is called an evolution operator. It describes the complete state of a system at a point t in time. The first axiom tells us that E (0 ) is the identity operator I. It gives the initial state at the time zero, the start, i.e. t = 0.
The second axiom implies that if the time is at a point s, and subsequently another amount of time t passes, the state of the space is the same as at the time s + t.
This is actually quite simple common sense stuff.

If an invertability axiom is added E becomes a dynamical system.
E ^{1} (t ) = E (t )
This third axiom, that for a dynamical system, depends on whether the operator E(t ) has an inverse, i.e. if time can be “reversed”. E(t ) would tell you the state of the system at a given time t before the time zero. That is why time would have to be historically eternal (as well as endless).
The concepts are really very simple and these axioms all seem trivial when you understand them. In fact these properties of time are logically and practically indispensable.
In an “eternal past” scenario you must have the last, the third axiom i.e. invertability. It means that the conditions for a dynamical systems are more restricted. In fact any dynamical system is also a semigroup since the semigroup axioms are satisfied, but not viceverca

However the question of whether time could have an end sounds like rather a strange idea. I'd think you would have to start off with saying what it is you mean.

Einstein's theories rejected the notion of simultaneity and the fact is intrinsic and unavoidable to the nature of time and space. As an example to say as I did, I am thinking of the universe as frozen at some instant in time like a snapshot. That is nonsense, strictly speaking. It is just the way I try to think about things. These days everybody knows about time dilation at incredibly high speeds, and occurring near bodies of immense mass such as socalled blackholes. Also the equivalence of mass and energy is common knowledge.
Special and General Relativity to me is just as strange and as much a mystery than to any man on street but I do understand enough to be convinced, and to know that there exist no inherent contradictions. Relativity theory is supported by much observational evidence and is thoroughly confirmed in experiments. I have spoken on different occasions to persons who I greatly respect and that I have total confidence in, who do know the mathematical theory and indeed, I am convinced. And yes, the nature of reality is radically different to our experience.
Philosophy
In essence a world of logic and thought consists of eternal and universal principles and is philosophy, and includes mathematics. Logic exists distinct of physical reality and experience. It is very surprising that it works at all and it does, and as far as we do know reality follows its unchanging rules and laws.
Not saying there are not disagreements but controversies are not inherent in the logic itself, but in the interpretation and in differences in opinion. The fact that someone believes something or not does not make that thing true or not. A statement is true or false, inherently.
Mathematics as such exists as eternal truths and therefore any new progress or original developments are not in any way invented, they are discovered. The truths are there and by themselves, whether known in the past at present or in future, or never.
“Plato is dear to me, but dearer still is truth.”
—Aristotle
∞ + ∞
From the Gospel According to St. John
ESV
In the beginning was the Word,
and the Word was with God,
and the Word was God.
In him was life, and the life
was the light of men.
The light shines in the darkness, and
the darkness has not overcome it.
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Comments
I tried hard to understand
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Nice work Tom, good luck
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