<p>Your assumption is not true. For example: if the probability density function of the atoms decreases quadratic exponentially (i.e. <span class="math-container">$e^{-r^2}$</span>), then their sum will be finite even in an infinite universe.</p>

<p>The sum of infinite many positive real numbers can be finite, although they need to decrease on a long-term.</p>

CC BY-SA 3.0
2016-09-09T15:41:28.317Z
@adrianmcmenamin Consider if each elemental particle in the Universe has a probability distribution with a non-zero value in the whole Universe. The sum of these wavefunctions will be not infinite anywhere, it shouldn't even had to be infinite in any point.
peterh
CC BY-SA 3.0
2014-07-03T13:31:01.587Z
Yes, this is true as well! A little counter-intuitive if you haven't come across Convergent Series (https://en.wikipedia.org/wiki/Convergent_series) before, but still true.
Takku
CC BY-SA 3.0
2014-07-04T14:42:38.493Z
yes, but what if we assume the Cosmological Principle applies and they are evenly distributed? Or do you mean the wave equation as it applies to each electron?
adrianmcmenamin
CC BY-SA 4.0
2014-07-03T11:45:28.883Z
2019-12-16T19:03:13.617Z
2019-12-16T19:03:13.617Z
peterh
<p>From a purely mathematical point of view; <a href="https://math.stackexchange.com/questions/36289/is-infinity-a-number">infinity is not a number</a>, use it as one at your peril.</p>

<p>Density is mass per volume, <code>mass / volume</code>. Infinite hydrogen atoms gives you infinite mass. An infinite universe gives you infinite volume. <a href="https://math.stackexchange.com/a/1406021/52721"><code>Infinity / Infinity</code> is indeterminate</a>.</p>

<p>But <code>mass / volume</code> is the formula for a <em>homogeneous</em> substance. The universe is not homogeneous, <em>it is heterogeneous</em>. The universe is different depending on where you look. Instead of a simple <code>mass / volume</code> you'd divide the universe into infinitesimally small volumes of space, measure the mass (and thus density) in each, and take the average.</p>

<p>This side-steps the infinity problem and you can get a sensible answer... assuming you can make infinite measurements.</p>

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2016-08-31T14:15:44.953Z
Obviously it's heterogeneous. That's the basis of the question.
adrianmcmenamin
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2016-08-31T17:32:54.133Z
@adrianmcmenamin Right. Yet you seem to be using the *homogeneous* formula for density. So there's your problem.
Schwern
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2016-08-31T06:31:26.240Z
2016-08-31T06:31:26.240Z
2017-04-13T12:19:35.350Z
Schwern
<p>I would like to point out that any information (even quantum mechanical) travels at most at the speed of light. (There are some problems with entangled systems etc, but we can assume most systems to not be entangled, which is a fair assumption, imo). So, if you consider the age of the universe to be finite, any point will have only finitely many overlapping wavefunctions (those within the range of visibility, and the probability can be very very small at most points). So that means that there is never an infinite amount of atoms in the visible region at any finite time, meaning that the density never reaches infinity (or does so only at t=inf).</p>

CC BY-SA 3.0
2014-07-02T09:19:58.030Z
2014-07-02T09:19:58.030Z
Takku
<p>If we make the assumption that the Universe is infinite, and has an infinite number of hydrogen atoms, then why is it not of infinite density - because, under Schrodinger's wave equation the probability of an electron being at any given point is non-zero and any non-zero number multiplied by infinity is itself infinity?</p>

<p>Is the answer (a) I have made some basic error in physics, (b) the Universe is provably not infinite because of this - effectively a version of Obler's Paradox or (c) the Pauli exclusion principle means that electrons just cannot be anywhere?</p>

CC BY-SA 3.0
2014-07-02T11:35:19.577Z
Having infinite number of hydrogen atoms is not a direct consequence of an infinite Universe. You must revise your hypothesis.
Py-ser
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2014-07-02T15:31:56.093Z
Who said it was a "direct consequence"?
adrianmcmenamin
CC BY-SA 3.0
2014-07-03T03:36:20.087Z
It seemed so from your question. Could you refer to the source which gave you these information?
Py-ser
<cosmology>
If the Universe is infinite, why isn't it of infinite density?
adrianmcmenamin
# If the Universe is infinite, why isn't it of infinite density?

(Asked by: username_0 on 2014-07-02)

If we make the assumption that the Universe is infinite, and has an infinite number of hydrogen atoms, then why is it not of infinite density - because, under Schrodinger's wave equation the probability of an electron being at any given point is non-zero and any non-zero number multiplied by infinity is itself infinity?

Is the answer (a) I have made some basic error in physics, (b) the Universe is provably not infinite because of this - effectively a version of Obler's Paradox or (c) the Pauli exclusion principle means that electrons just cannot be anywhere?

**Tags:** cosmology

> **Comments:**
> **username_1**: Having infinite number of hydrogen atoms is not a direct consequence of an infinite Universe. You must revise your hypothesis.  
> **username_0**: Who said it was a "direct consequence"?  
> **username_1**: It seemed so from your question. Could you refer to the source which gave you these information?  

---

## 3 answers

### Answer by username_3 on 2014-07-02. Score: 7 | Accepted answer

I would like to point out that any information (even quantum mechanical) travels at most at the speed of light. (There are some problems with entangled systems etc, but we can assume most systems to not be entangled, which is a fair assumption, imo). So, if you consider the age of the universe to be finite, any point will have only finitely many overlapping wavefunctions (those within the range of visibility, and the probability can be very very small at most points). So that means that there is never an infinite amount of atoms in the visible region at any finite time, meaning that the density never reaches infinity (or does so only at t=inf).


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### Answer by username_2 on 2014-07-03. Score: 6

Your assumption is not true. For example: if the probability density function of the atoms decreases quadratic exponentially (i.e. $e^{-r^2}$), then their sum will be finite even in an infinite universe.

The sum of infinite many positive real numbers can be finite, although they need to decrease on a long-term.

> **Comments:**
> **username_3**: Yes, this is true as well! A little counter-intuitive if you haven't come across Convergent Series (https://en.wikipedia.org/wiki/Convergent_series) before, but still true.  
> **username_0**: yes, but what if we assume the Cosmological Principle applies and they are evenly distributed? Or do you mean the wave equation as it applies to each electron?  
> **username_2**: @username_0 Consider if each elemental particle in the Universe has a probability distribution with a non-zero value in the whole Universe. The sum of these wavefunctions will be not infinite anywhere, it shouldn't even had to be infinite in any point.  

---

### Answer by username_4 on 2016-08-31. Score: 3

From a purely mathematical point of view; [infinity is not a number](https://math.stackexchange.com/questions/36289/is-infinity-a-number), use it as one at your peril.

Density is mass per volume, `mass / volume`. Infinite hydrogen atoms gives you infinite mass. An infinite universe gives you infinite volume. [`Infinity / Infinity` is indeterminate](https://math.stackexchange.com/a/1406021/52721).

But `mass / volume` is the formula for a *homogeneous* substance. The universe is not homogeneous, *it is heterogeneous*. The universe is different depending on where you look. Instead of a simple `mass / volume` you'd divide the universe into infinitesimally small volumes of space, measure the mass (and thus density) in each, and take the average.

This side-steps the infinity problem and you can get a sensible answer... assuming you can make infinite measurements.

> **Comments:**
> **username_0**: Obviously it's heterogeneous. That's the basis of the question.  
> **username_4**: @username_0 Right. Yet you seem to be using the *homogeneous* formula for density. So there's your problem.  

---

