# Where does idea of infinite size of universe come from?

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I'm just wondering how did we start to think about possibility of infinite size of the universe?

I know that positive curvature implies bounded universe, while zero and negative curvature can be both bounded and unbounded (specifically if we abandon idea of isotropy on very large scales), my question here is why do we consider infinite (unbounded) universe as a possibility at all?

No other physical quantity has ever been measured or perceived as infinite, and some previously thought as infinite were found to be finite after all (speed of light). There are simply no infinite things in universe, and infinity is just a product of our mind, our abstractions. Even definition of something infinitely large is somewhat lacking, and again based on abstraction - we need a set with infinitely many members to define infinitely large thing, no finite set will support definition, as it will actually be supremum of that set (larger than any other member of the set).

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I think it is a combination of factors.

We are able to model the cosmos well on the assumption of a flat simple topology without a boundary and on large scales isotropic in content.

This leads to a spatially flat universe from t = 0 to today.

It maybe that we can't see enough of the Universe to detect deviations. However many possible topologies and curvatures are ruled out by observation.

Cantor put infinity on a firm mathematical footing defining different types although it drove him mad. I see infinity as just as well defined as any continuum measurement which is just as problematic in physics.

Regards Andrew

Edited by andrew s
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I think it just adds unnecessary complexity thus violating Occam's razor.

If we assume infinite universe, than it is justified to talk about Multiverse, or rather there exists observable patch that contains every sequence of events possible - and what's more - there are infinite number of such observable patches with exact same settings.

There are infinite number of copies of me writing this post, both "now", a second ago, two seconds ago, ...., but also "future me" doing the same in 1s future, 2s, 3s, and not to mention infinite time increments

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1 minute ago, vlaiv said:

I think it just adds unnecessary complexity thus violating Occam's razor.

If we assume infinite universe, than it is justified to talk about Multiverse, or rather there exists observable patch that contains every sequence of events possible - and what's more - there are infinite number of such observable patches with exact same settings.

There are infinite number of copies of me writing this post, both "now", a second ago, two seconds ago, ...., but also "future me" doing the same in 1s future, 2s, 3s, and not to mention infinite time increments

I don't agree with these conclusions.  Can you define a finite Universe that fits our current understanding without bringing issues of its own?

Recall space time is not infinite, it is only spatially finite. Space time is geometry.

Are you willing to accept there are an infinite number of points on a line? And following that if you dive the line (however unevenly) both segments have the same infinite number of points as the first and each other? Just as problematic ?

Regards Andrew

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Defining finite Universe that fits both our current understanding and current measurements is easy - 3 sphere for example with curvature close to zero and within current measurement error. It is positive curvature geometry, with lower bound on radius - from curvature measurements, but finite in extent.

I don't follow second part of your post. I have no problem with abstract thinking of infinity - I do have a problem with assigning infinity to anything physical.

To expand on that, and use line / line segment analogy - we can assume number line and detected particle position as being real number line with infinite numbers - but is it really so? Very quickly we run out of comparison sizes when we try to measure more precisely distance. Take Planck length - it is about 10^-35. That is seriously "grained" compared to infinities - even 10^(10^(10^(10^....)))))^-35 - power repeated 10^35 is finitely huge number compared to infinitesimally small (I see definition of infinity a la Douglas Adams).

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2 hours ago, vlaiv said:

Defining finite Universe that fits both our current understanding and current measurements is easy - 3 sphere for example with curvature close to zero and within current measurement error. It is positive curvature geometry, with lower bound on radius - from curvature measurements, but finite in extent.

I don't follow second part of your post. I have no problem with abstract thinking of infinity - I do have a problem with assigning infinity to anything physical.

But, its finite extent notwithstanding,  a 3-sphere has the same cardinality as the real line, and thus has same "fault" as a model of positions in space as (any segment of) the real line ...

2 hours ago, vlaiv said:

To expand on that, and use line / line segment analogy - we can assume number line and detected particle position as being real number line with infinite numbers - but is it really so? Very quickly we run out of comparison sizes when we try to measure more precisely distance. Take Planck length - it is about 10^-35. That is seriously "grained" compared to infinities - even 10^(10^(10^(10^....)))))^-35 - power repeated 10^35 is finitely huge number compared to infinitesimally small (I see definition of infinity a la Douglas Adams).

Edited by George Jones
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I probably did not make my self clear enough, so I'll expand on above (and maybe try to be more precise).

Argument about line segment, context of set members infinity - infinite real numbers between 0 and 1 for example has to do with my assertion that we don't encounter infinities in physical world. One can naively assume that for example position coordinate can be any real number - suppose we have electron somewhere along the 1 meter ruler and we can assign any real number to its position - this would represent infinity in physical world. However we are not able to do so - measurement is comparison of quantities - we need something small enough to say this many such small units there are between start of ruler and detected electron position. There is however lower bound on meaningful length - we cannot produce any physical object smaller than this bound to make measurement. If we tried for example to make photon with wavelength sufficiently small, it would be so energetic that it would collapse into tiny black hole.

Even if we did not have this restriction, and could in principle use all remaining matter/energy in our causally connected patch of the universe to form a single photon - it would still have upper bound on amount of energy and thus minimum wavelength. Smaller lengths than this simply don't have physical meaning in similar (or even same sense) as Heisenberg uncertainty principle - there is intrinsic limit to measurement that is part of physical reality.

Same argument goes for time - one can't measure arbitrary short interval of time - for the same reason as above + limit on speed of causality. In order to have comparison "clock" - it needs to "tick" and we identify tick as changed internal state, but there is limit how precise positions can be defined and also limit in how fast causality travels - we won't be able to distinguish two different states of clock system for short enough time.

Back to original infinity. Since we assume following: space time is flat, isotropic and homogeneous - only geometry of space-time with those properties is unbounded. This is what we call infinite universe - meaning there is no upper bound on distance between two arbitrary points.

This leads to all kind of weird stuff that we can conclude - my point about Multiverse - fact that in space time of infinite extent there are infinitely many "observable universes", and among those infinite number of "observable universes" - every possible configuration will exist, and not only that - it will be repeated infinitely many times.

Then Andrew asked me to point out to geometry that is compatible with our current understanding and also observations and measurements that is bounded - meaning there is "distance" such that distance between any two points in geometry is less than this.

If we want to step down in number of dimensions - think straight line and circle. We could say that circle is infinite - one can go around the circle infinite number of times (but again not - subject of age of universe and limit to max speed), but distance along the circle between any two points on it will have upper bound - circle is bounded.

In same sense 3 sphere is bounded space, but it's positively curved space. Our measurements show that space is flat with 2% uncertainty - that means that curvature can be positive small value - like 3 sphere with large radius of curvature.

It can also mean that curvature is negative, or that it is indeed flat if curvature is precisely 0 (most problematic case for measurement).

Mind you there are other possibilities that are compatible with our observations so far - for example we live in flat geometry that is bounded. Example of this would be 3 torus. However in that case we would have problem with isotropy - or rather measured isotropy can be used to put lower bound on "size" of universe.

My original question was - why do we subscribe to "simplest" case - flat infinite universe when infinite part goes against all our understanding of the nature?

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I tried using "A brief history of time", by some guy who wrote a couple of books on physics.

After the 5th reading, only on the 2nd, I may start to understand the first chapter.

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3 hours ago, vlaiv said:

My original question was - why do we subscribe to "simplest" case - flat infinite universe when infinite part goes against all our understanding of the nature?

My view is that it does not go against all our understanding of nature.

Your view on multiple copies of self, universes etc. is in my view mistaken. If you have an infinite number of trials then possibly you view could be true. However, we have just one universe (one instance) and in this case there is just one outcome. One you, one me, there is no statistical reason for replication.

On a sphere the coming back on yourself could have had specific effects when the scale factor was small in the early universe before  casually disconnection. I suspect this should be have left an imprint on the CMB. I think this will have been looked for.

Regards Andrew

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I've always treated infinity as an abstraction. To my understanding it does not exist in the physical world rather it's an annoyance that turns up in our equations when we have a less than complete understanding.   I can't help but think like Cantor we just create models of infinity that would ultimately drive us crazy.  Take for example a universe that has a fixed size, a measurable distance from one end to the other.  By definition, being the largest dimension available we could argue that that represents infinity ( nothing can be larger than the universe). However that is unsatisfactory as infinity by definition cannot be bound.   I'm happy for infinity to exist in mathematics where it serves a purpose in making our equations work. For it to exist in reality, I would find that difficult if not impossible to accept.

The cartoon below I guess sums up my approach to infinity

Jim

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4 minutes ago, andrew s said:

Your view on multiple copies of self, universes etc. is in my view mistaken. If you have an infinite number of trials then possibly you view could be true. However, we have just one universe (one instance) and in this case there is just one outcome. One you, one me, there is no statistical reason for replication.

You in fact have a infinite number of trials by virtue of infinite spatial extent. We will have infinite number of "observable spheres" that evolved from same big bang, and if we assume homogeneity, all those "spheres" are in principle "the same" - here not meaning they are identical copies, but rather governed by same laws and having similar initial conditions. Once you have infinite amount of something - no matter how complex system, it is bound to repeat it self.

Think of it this way - take all real intervals 0-1, 1-2, 2-3, .... up to infinity

and from each interval choose any real number and observe decimal digits (just digits after decimal place - we are not interested in number in front) - as a sequence of numbers. It does not matter if such sequence is finite or infinite.

What is the probability of having the same sequence show up twice or more times?

Fact that space is infinite means that all of these occur "simultaneously".

In fact, have a look at 4 levels of multiverse by Max Tegmark:

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#### Level I: An extension of our Universe

A prediction of chaotic inflation is the existence of an infinite ergodic universe, which, being infinite, must contain Hubble volumes realizing all initial conditions.

Accordingly, an infinite universe will contain an infinite number of Hubble volumes, all having the same physical laws and physical constants. In regard to configurations such as the distribution of matter, almost all will differ from our Hubble volume. However, because there are infinitely many, far beyond the cosmological horizon, there will eventually be Hubble volumes with similar, and even identical, configurations. Tegmark estimates that an identical volume to ours should be about 1010115 meters away from us.[26]

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.[59] This follows directly from the cosmological principle, wherein it is assumed that our Hubble volume is not special or unique.

Actually, I completely agree with level 4 - representing all possible universes - and not only that - showing that they are in fact the same thing.

Hugh Everett's many worlds interpretation is "mathematically" identical to this infinite copies thing - all outcomes will be a) realized in some of universes that had completely the same history up to "split" b) there will be infinite number of universes with any particular outcome ensuring that further splits again realize every possibility.

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7 hours ago, andrew s said:

Are you willing to accept there are an infinite number of points on a line? And following that if you dive the line (however unevenly) both segments have the same infinite number of points as the first and each other? Just as problematic ?

Regards Andrew

You can divide an imaginary line an infinite number of times but not a real one.

It is dangerous to make generalised physical assumptions that at first glance seem obvious but on inspection are less so.

For example..... a real line can be divided into an infinite number of pieces....this seems common sense but upon inspection we realise that distance is a function of time.

Consider measuring a length of a line A to B, whatever measurement system you use it takes an amount of time to say "point A is here" and then move to point B ( not just physically but abstractly as well ) and then say "point B is here" the time taken between these two measurement ( or even just statements ) cannot be zero. Therefore this suggests that there is a Finite smallest measurement.

A real world example of this would be to say that to get from point A to point B you first half to get halfway and to get halfway you have to get a quarter of the way. This would go on for ever and it would be impossible to move in any direction.

We know this by observation to be untrue therefore this again proves that there is a Finite smallest length and a Finite smallest amount of time.

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Also by observation the Universe is clearly not infinite...

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3 minutes ago, Kev M said:

Also by observation the Universe is clearly not infinite...

The observable Universe is finite but we have no way of knowing what lies beyond that.

Alan

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@vlaiv we will just have to disagree on this. I don't accept that dividing up one universe leads to multiple trials. I basically reject the logic that is commonly applied in arguments along these lines.

I respect your view but disagree. I don't have the motivation to go further into this.

Regards Andrew

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17 minutes ago, Kev M said:

Also by observation the Universe is clearly not infinite...

Can you provide examples?

Regards Andrew

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19 minutes ago, Kev M said:

We know this by observation to be untrue therefore this again proves that there is a Finite smallest length and a Finite smallest amount of time.

Strange then that the two most successful physical theories Quantun Field Theory and General Relativity both are based on a continuous spacetime.

Regards Andrew

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32 minutes ago, Kev M said:

A real world example of this would be to say that to get from point A to point B you first half to get halfway and to get halfway you have to get a quarter of the way. This would go on for ever and it would be impossible to move in any direction.

This is one of Zeno's paradoxes.  It was solved by calculus and the proof that infinite series can converge to a finite number. Simply the increasing number of half steps is balanced by the decreasing time it takes to traverse them.

Regards Andrew

Edited by andrew s
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47 minutes ago, saac said:

I've always treated infinity as an abstraction. To my understanding it does not exist in the physical world rather it's an annoyance that turns up in our equations when we have a less than complete understanding.   I can't help but think like Cantor we just create models of infinity that would ultimately drive us crazy.  Take for example a universe that has a fixed size, a measurable distance from one end to the other.  By definition, being the largest dimension available we could argue that that represents infinity ( nothing can be larger than the universe). However that is unsatisfactory as infinity by definition cannot be bound.   I'm happy for infinity to exist in mathematics where it serves a purpose in making our equations work. For it to exist in reality, I would find that difficult if not impossible to accept.

The cartoon below I guess sums up my approach to infinity

Jim

So what about the key constants pi, e which are transcendental with an infinite number of digits.  Is the area of a unit circle not real? Is it truncated as there are more digits in pi than atoms in the observable universe?

Regards Andrew

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2 minutes ago, andrew s said:

So what about the key constants pi, e which are transcendental with an infinite number of digits.  Is the area of a unit circle not real? Is it truncated as there are more digits in pi than atoms in the observable universe?

Regards Andrew

Area of unit circle is as much real as circle itself.

If you can make true physical circle - meaning composed of "all points" in plane equally distant from origin then that circle will indeed have area of a number written in decimal system with infinite number of digits.

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6 minutes ago, andrew s said:

So what about the key constants pi, e which are transcendental with an infinite number of digits.  Is the area of a unit circle not real? Is it truncated as there are more digits in pi than atoms in the observable universe?

Regards Andrew

As a novice all that implies is that our understanding of Maths be it an invented or discovered concept has a long way to go and only partially represents what realy going on.

Alan

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I think @andrew s has a good point about e and pi, but that makes me wonder.

Both of these constants are in fact infinite sums.

Does any one know any physics equation where replacement of such constants with finite sum would yield significantly different result other than discrepancy below threshold of measurement? After all - we never use true value of any of given constants in our calculations - we always have to settle for approximation (finite precision), otherwise it would take infinity long time to perform calculation.

However, there might be mathematical identity that is only true for infinite sums and diverges considerably when finite version is used, and such identity used to derive physical law.

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2 minutes ago, Alien 13 said:

As a novice all that implies is that our understanding of Maths be it an invented or discovered concept has a long way to go and only partially represents what realy going on.

Alan

No one has any idea of what is really going on. All we have are models more or less useful at predicting what we observe as goiog on.

We could all be in a simulation run by transdimentional mice or the matrix.

Personally I take a realist stance and believe in an external reality. However,  I don't know what anything is in reality and challenge anyone to give an example of what is really real.

Regards Andrew

PS my last comment on this.

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22 minutes ago, vlaiv said:

However, there might be mathematical identity that is only true for infinite sums and diverges considerably when finite version is used, and such identity used to derive physical law.

Ok one more.

If you compare Bose Einstein and Fermi Dirac statistics if e were not exactly e then the +/- 1 would not lead to the divergent behaviour of fermions and bosons but something less  dramatically different.

The end from me.

Regards Andrew

Forget the above it's wrong any number to power zero equals one.

Opps

Edited by andrew s
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9 hours ago, vlaiv said:

I think it just adds unnecessary complexity thus violating Occam's razor.

I would say that adding a boundary to the Universe adds complexity. What does it even mean for a universe to have a boundary?

There is no evidence for a boundary anywhere that we can observe - surely that is where Occam's razor should be applied.

Edited by Gfamily

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