If the universe is truly infinite in size....

[abbott287]

would that mean the Big Bang never happened? If not, please explain to the best of your ability as to why. Much appreciated.

 

[mfb]

would that mean the Big Bang never happened?

No.

The universe could have started at an infinite size. There is no problem with that model. If it is infinite now it was infinite as long as it existed. We don't know if it is.

 

[russ_watters]

would that mean the Big Bang never happened? If not, please explain to the best of your ability as to why. Much appreciated.

Being infinite does not preclude expansion and contraction. If you draw dots on a rubber band and pull it apart, the dots get further apart. It doesn't matter how long the rubber band is.

 

[Arman777]

First thing you should notice or consider that big bang did not happened at a point but it happened everywhere in the universe.

Think an infinite size of paper and in this paper you are using grid coordinates. Let's suppose you choosed the distance between every point on this grid to be $D$.

While we go back in time, this distance ($D$) becomes smaller and smaller. Now let's go back in time, 0.000000000000001 seconds after the big bang. At this moment the distance between two points will be very small but universe would be still infinite.

Where at the big bang the distance was $0$ between two points, but also universe was infinite in size. Its hard to imagine yes, but its also what we mean by "big bang happened everywhere", cause all (infinite) universe was at that point while the distance was 0 between each point, which we call that point singularity.

 

[jbriggs444]

Where at the big bang the distance was $0$ between two points, but also universe was infinite in size. Its hard to imagine yes, but its also what we mean by "big bang happened everywhere", cause all (infinite) universe was at that point while the distance was 0 between each point, which we call that point singularity.

You cannot extrapolate all the way back to the singularity. No coordinate chart for any portion of the universe goes that far. That's part of what it means to be a "singularity".

A singularity is not a point.

 

[Arman777]

You cannot extrapolate all the way back to the singularity. No coordinate chart for any portion of the universe goes that far. That's part of what it means to be a "singularity".

A singularity is not a point.

I see

 

[Grinkle]

would that mean the Big Bang never happened? If not, please explain to the best of your ability as to why. Much appreciated.

The following is, at best, a "B" response to your "I" thread.

The switch you might make is to think of the universe as getting denser and denser as you roll the clock back - NOT smaller and smaller. Then perhaps you ask what is the universe expanding into if it was always infinitely large and its now somehow getting less dense without losing any particles, which is a different visualization problem, but at least its one that is more aligned with what expansion theories are saying about the past universe.

My own thinking on this is not at all visual. Infinities need not be of the same count. There are an infinite count of odd integers, and an infinite count of even integers, and obviously the count of all integers is twice that of either only odds or only evens. That doesn't help me visualize anything, but it helps me get to accepting that something being infinite does not mean it cannot possibly still be "more" than it is.

 

[PAllen]

A visualization I found useful first grappling with these ideas is to imagine an infinite space at a moment in cosmological time as being chopped into one inch cubes. You have a (countably) infinite collection of these cubes. A bit later, all the cubes are two inches on a side. They still just assemble into an infinite space. Now start compressing then to smaller cubes. No matter how many time you reduce all the cube sizes by half, they still assemble into an infinite space. As noted by @jbriggs444, you cannot take this process all the way back to zero size cubes.

For me, at the beginning, this notion of a magical infinite bag of cubes that grow or shrink, made the whole thing more mentally palatable. It sidesteps the mental block of what is the expansion into, and helps pull away from the idea of ‘how does everything grow without getting in each other’s way.

 

[mfb]

My own thinking on this is not at all visual. Infinities need not be of the same count. There are an infinite count of odd integers, and an infinite count of even integers, and obviously the count of all integers is twice that of either only odds or only evens. That doesn't help me visualize anything, but it helps me get to accepting that something being infinite does not mean it cannot possibly still be "more" than it is.

There are as many odd integers as there are integers. Proof: There is a one-on-one mapping between them (e.g. 1 to 1, 2 to 3, 3 to 5, 4 to 7, ...) I don't think this helps here.

 

[PeroK]

A visualization I found useful first grappling with these ideas is to imagine an infinite space at a moment in cosmological time as being chopped into one inch cubes. You have a (countably) infinite collection of these cubes. A bit later, all the cubes are two inches on a side. They still just assemble into an infinite space. Now start compressing then to smaller cubes. No matter how many time you reduce all the cube sizes by half, they still assemble into an infinite space. As noted by @jbriggs444, you cannot take this process all the way back to zero size cubes.

That's just Zeno's paradox in another form. Imagining that for something to get to $0$ it must halve an infinite number of times and can never get there.

Mathematically, you can easily have a (distance) function that continuously reduces to $0$ in finite time. For example, for a matter dominated universe the function could be $a(t) = (t/t_0)^{2/3}$. That function, mathematically, quite happily goes to $0$ at $t = 0$. Even more simply the linear function $a(t) = t$ would do the same.

At $t=0$ the distance between any two points would be $0$. Mathematically that's not an issue, per se. At every time except $t=0$ you have a valid metric and at $t=0$ the metric is gone!

The real issue is the physical interpretation of this: which might be that distance as a physically measurable quantity has ceased to exist. Could you could interpret that as that "space" has ceased to exist? Also, as the distance between any two points reduces towards $0$, the density increases without bound. And, at $t=0$, the density is either "infinite" or more precisely "undefined".

Most people seem to visualise the expansion as leading back to a single point. This appeals to the physical and mathematical notion that:

$x = y$ if and only the distance between $x$ and $y$ is $0$. In other words, if the distance between any two points is zero, then you have only one point.

But, mathematically, if you consider $\mathbb{R}^3$, say, without a well-defined metric, then it's just the same old infinite set of points but without the concept of distance.

Another way to visualise the singularity, therefore, is to imagine the underlying set of points staying exactly where they are, but this thing we call and measure distance reduces until at $t=0$ the concept of distance itself has gone. And, the problem is that we have no description of the laws of physics that would support this process all the way back to $t=0$.

 

[PAllen]

That's just Zeno's paradox in another form. Imagining that for something to get to $0$ it must halve an infinite number of times and can never get there.

Mathematically, you can easily have a (distance) function that continuously reduces to $0$ in finite time. For example, for a matter dominated universe the function could be $a(t) = (t/t_0)^{2/3}$. That function, mathematically, quite happily goes to $0$ at $t = 0$. Even more simply the linear function $a(t) = t$ would do the same.

At $t=0$ the distance between any two points would be $0$. Mathematically that's not an issue, per se. At every time except $t=0$ you have a valid metric and at $t=0$ the metric is gone!

The real issue is the physical interpretation of this: which might be that distance as a physically measurable quantity has ceased to exist. Could you could interpret that as that "space" has ceased to exist? Also, as the distance between any two points reduces towards $0$, the density increases without bound. And, at $t=0$, the density is either "infinite" or more precisely "undefined".

Most people seem to visualise the expansion as leading back to a single point. This appeals to the physical and mathematical notion that:

$x = y$ if and only the distance between $x$ and $y$ is $0$. In other words, if the distance between any two points is zero, then you have only one point.

But, mathematically, if you consider $\mathbb{R}^3$, say, without a well-defined metric, then it's just the same old infinite set of points but without the concept of distance.

Another way to visualise the singularity, therefore, is to imagine the underlying set of points staying exactly where they are, but this thing we call and measure distance reduces until at $t=0$ the concept of distance itself has gone. And, the problem is that we have no description of the laws of physics that would support this process all the way back to $t=0$.

If you go to zero, you have a countably infinite set of points, which is no linger R3 or any form of continuum. Thus a singular change has occurred. This has nothing to zeno’s paradox. The suggested cutting in half is just a verbal aid. It doesn’t matter what function you use.

 

[Arman777]

You cannot extrapolate all the way back to the singularity. No coordinate chart for any portion of the universe goes that far. That's part of what it means to be a "singularity".

A singularity is not a point.

If you go to zero, you have a countably infinite set of points, whichis no linger R3 or any form of continuum. Thus a singular change has occurred.

Can you go to zero or not. I am confused ?

Cause " a singular change has occured" is this happens at t=0 ?

 

[PAllen]

Can you go to zero or not. I am confused ?

No, you can’t. The hypothetical zero state cannot be part of the same manifold, precisely because it can no longer be homeomorphic to a continuum.

 

[PeroK]

Can you go to zero or not. I am confused ?

Cause " a singular change has occured" is this happens at t=0 ?

The state at $t=0$ has a problem. Whether you think of it as a single point or a distance-less universe, it's not something that we have any sensible way of analysing (as a universe). Both of those options represent singular changes.

You cannot map the universe to either a single point or a distance-less set of points in any meaningful way. That is a singularity (in the model of the universe we have).

 

[Arman777]

I understand now thanks

 

[PeroK]

If you go to zero, you have a countably infinite set of points, which is no linger R3 or any form of continuum. Thus a singular change has occurred. This has nothing to zeno’s paradox. The suggested cutting in half is just a verbal aid. It doesn’t matter what function you use.

Actually, Zeno would be unperturbed by the Big Bang singularity. He would simply argue:

We are at $t_0$. At some earlier time we were at $t_0/2$ and at some earlier time at $t_0/4$. No matter how many times we halve the time, we never get to $t=0$; therefore, $t=0$ never happened and the problem of the singularity is solved!

 

[PAllen]

Actually, Zeno would be unperturbed by the Big Bang singularity. He would simply argue:

We are at $t_0$. At some earlier time we were at $t_0/2$ and at some earlier time at $t_0/4$. No matter how many times we halve the time, we never get to $t=0$; therefore, $t=0$ never happened and the problem of the singularity is solved!

But that isn’t the intent of my argument. It wasn’t that you could not reach t=0 , it was the the t=0 state can no longer be assembled into a continuum because the cardinality is wrong.

 

[PeroK]

But that isn’t the intent of my argument. It wasn’t that you could not reach t=0 , it was the the t=0 state can no longer be assembled into a continuum because the cardinality is wrong.

The cardinality needn't change. At all times, including $t=0$, you can have the full uncountable infinity of points in space. The only thing that need change with time is the measure of distance between any two points.

At $t=0$ you either reduce to single point - which is still a valid metric/topological space - but can't be homeomorphic to the current universe. Or, you reduce to a set without a metric, which, by definition, can't be homeomorphic to anything.

 

[PAllen]

The cardinality needn't change. At all times, including $t=0$, you can have the full uncountable infinity of points in space. The only thing that need change with time is the measure of distance between any two points.

At $t=0$ you either reduce to single point - which is still a valid metric/topological space - but can't be homeomorphic to the current universe. Or, you reduce to a set without a metric, which, by definition, can't be homeomorphic to anything.

Homeomorphism does not require a metric. It depends only on open set structure.

I guess the crux of the issue is whether you can validly assign a zero measure to an open ball of R3. So far as I understand measure theory (which isn't that deep), the answer is no.

 

[PeroK]

Homeomorphism does not require a metric. It depends only on open set structure.

Okay, a set without a topology if you prefer. Mathematically, you don't have to have such things. $\mathbb{R}^3$ is perfectly happy mathematically as a just set of points.

On a lighter note, if you were asked to conjure a universe, you might be better off starting with an infinite set of points without any defined topological structure, than starting with a single point, which trivially has a topological structure.

 

[Arman777]

I don't understand the math but it might be helpfull,
https://physics.stackexchange.com/questions/151027/interpretation-of-a-singular-metric

 

[PAllen]

Okay, a set without a topology if you prefer. Mathematically, you don't have to have such things. $\mathbb{R}^3$ is perfectly happy mathematically as a just set of points.

On a lighter note, if you were asked to conjure a universe, you might be better off starting with an infinite set of points without any defined topological structure, than starting with a single point, which trivially has a topological structure.

The two requirements: volume of zero by some valid measure and homeomorphic to R3 are contradictory. Thus, a zero volume state cannot be part of the same topological space as any nonzero volume. That is the mathematical statement of "the initial state cannot be part of the universe"

 

[PeroK]

The two requirements: volume of zero by some valid measure and homeomorphic to R3 are contradictory. Thus, a zero volume state cannot be part of the same topological space as any nonzero volume. That is the mathematical statement of the initial state is not part of the universe.

Or, by definition, the initial state is part of the universe and the assumptions about the mathematics that govern its evolution must be wrong in some way.

 

[Grinkle]

There are as many odd integers as there are integers. Proof: There is a one-on-one mapping between them (e.g. 1 to 1, 2 to 3, 3 to 5, 4 to 7, ...) I don't think this helps here.

Doh! Thanks, I am corrected.

 

[frantkfoe]

Can there really be an infinity of things? An infinity of numbers, for example. What do we mean when we say we can count from one to infinity? Well we certainly can't. It would take an infinity to do that. We can imagine that there are infinite numbers, an infinite number of numbers, if you will. But there can't be a number that we can point to and say “that's the infinite number”. Because in that case there would always be a finite number of numbers. And if X is the infinite number what about X +1? We can imagine an infinite succession of numbers n+1 without end. But again there would always be a finite number of numbers at any given point.

In the same sense and infinite space could never be realized. An infinite space cannot be measurable. If it was measurable it would have a boundary and therefore would be finite. And what would lie beyond that boundary? More space? If you have a space that's infinitely expanding it would always be a finite space at any given time.

If there was an infinity of time then time would not have had a beginning. If time had no beginning it could never have begun. There could never be any progression of time because there would always be an infinite regression that could never reach a starting point.

 

[jbriggs444]

Can there really be an infinity of things? An infinity of numbers, for example.

All of these things are worked out in modern (post-Cantorian) mathematics. We can speak of sets with infinitely many things without any of those things being infinite. But further discussion of that probably belongs over in the Mathematics forum.

 

[PeterDonis]

Or, by definition, the initial state is part of the universe and the assumptions about the mathematics that govern its evolution must be wrong in some way.

In which case we are no longer talking about an actual scientific model, but just speculating about some other hypothetical model that doesn't exist. Which is off topic for PF discussion. @PAllen is describing the actual scientific model that is our current best fit for our universe. That is what is on topic for this discussion.

 

[junjunjun233]

.

If there was an infinity of time then time would not have had a beginning. If time had no beginning it could never have begun. There could never be any progression of time because there would always be an infinite regression that could never reach a starting point.

Math has a lot of avenues to do that. We can only extrapolate from observation. It is impossible to know for sure if the universe is truly infinite. The observable universe is definitely finite. Interestingly enough, time is more intriguing than infinity itself.^^

 

[abbott287]

First thing you should notice or consider that big bang did not happened at a point but it happened everywhere in the universe.

Think an infinite size of paper and in this paper you are using grid coordinates. Let's suppose you choosed the distance between every point on this grid to be $D$.

While we go back in time, this distance ($D$) becomes smaller and smaller. Now let's go back in time, 0.000000000000001 seconds after the big bang. At this moment the distance between two points will be very small but universe would be still infinite.

Where at the big bang the distance was $0$ between two points, but also universe was infinite in size. Its hard to imagine yes, but its also what we mean by "big bang happened everywhere", cause all (infinite) universe was at that point while the distance was 0 between each point, which we call that point singularity.

You seem to be contradicting yourself. You said do not think of it as a point, which is how I think of it. Then you told me the distance beteeen points was zero, which makes everything a point! Then if the big bang happened everywhere, there was a lot of something there pre big bang!

 

[abbott287]

The following is, at best, a "B" response to your "I" thread.

The switch you might make is to think of the universe as getting denser and denser as you roll the clock back - NOT smaller and smaller. Then perhaps you ask what is the universe expanding into if it was always infinitely large and its now somehow getting less dense without losing any particles, which is a different visualization problem, but at least its one that is more aligned with what expansion theories are saying about the past universe.

My own thinking on this is not at all visual. Infinities need not be of the same count. There are an infinite count of odd integers, and an infinite count of even integers, and obviously the count of all integers is twice that of either only odds or only evens. That doesn't help me visualize anything, but it helps me get to accepting that something being infinite does not mean it cannot possibly still be "more" than it is.

This is why an infinite may not be possible. Only in theory. My infinite would be truly infinite. If you had an infininte space with infinite anything in it, it would be completely filled with that one thing. But I see how you could play the game both ways and make sense, which is why I don't believe there is anything truly infinite at this time.

 

[jbriggs444]

This is why an infinite may not be possible. Only in theory. My infinite would be truly infinite. If you had an infininte space with infinite anything in it, it would be completely filled with that one thing. But I see how you could play the game both ways and make sense, which is why I don't believe there is anything truly infinite at this time.

Naive intuition is no substitute for a course in real analysis.

 

[abbott287]

Being infinite does not preclude expansion and contraction. If you draw dots on a rubber band and pull it apart, the dots get further apart. It doesn't matter how long the rubber band is.

That rubber band would have to be stretching into something, so the universe could not be infinite in that model.

 

[abbott287]

Naive intuition is no substitute for a course in real analysis.

Define infinite.

 

[jbriggs444]

That rubber band would have to be stretching into something

That turns out not to be the case. One can describe a notional rubber band without requiring that it be embedded in a higher dimensional space, infinite or not.

 

[jbriggs444]

Define infinite.

One definition is "a set is infinite if there exists a bijection between a proper subset and the whole set". That's the Dedekind definition.