A trip to infinity

Introduction

Have you ever lied down watching the stars and wondered what is there beyond the stars, beyond what our eyes could see? If you’re like me you’d have thought that universe extends towards infinity. It is never ending.

Infintity is confusing

But yesterday I watched a show on Netflix called - A trip to infinity which ended up blowing my assumptions.
They started by introducing the mathematical concept of infinity- where you an always add 1 and get a bigger number. There are some concepts that don’t fit neatly with the arithmetic rules.

For e.g.

\(\infty + \infty = \infty \\ \)

And if you subtract two infinities you get 0

\( \infty - \infty = 0\)

If you subtract infinity on both sides of equation 1 you get a unintuitive result

\( \infty = 0\)

What does that mean?

Larger Infinity

What that result shows is that not all infinite sets are the same. You can compare infinite sets by showing one to one corresponce.
For e.g. Consider a set like 0.1, 0.2 etc. You can show a correspondence of

0.1 - 1

0.2 - 2

0.3 - 3 etc.. to show that they correspond 1:1 and hence these sets are equal

But there are other numbers like

\(\sqrt{2}, \pi, e\)

are different kind of infinity. If you keep writing out the numbers of pi, you can keep creating an infinite set. This set cannot be mapped to the counting infinity set of 1,2,3, etc. Hence this is a different infinte set from the counting infinity set and is considered larger because it is not countable.

Apple in a box

This was the most interesting mind-blowing concept from the documentary. It goes like this - If you have an apple in a box from which nothing can escape and after a long time the apple gets rotten, in even more time, it turns into a gas and because of the chemical energy, the gas gets heated, then there will be a nuclear reaction forming plasma and subatomic particles and then slowly form matter. An apple has approximated 10^23 atoms. So they argue that there are only a finite states of matter and given enough time matter will cycle through all states and we might get to the original apple again. What that means is that everything in the world will form again given enough time.
The box is the universe and we are the apples. In that case, we will exist again. And if there are multiple universes there might be other people like us now. We may never meet them because those universes are far way and we cannot go faster than the speed of light.

This is called the poincare recurrence. This is called a ergodic hypothesis which assumes every microstate of the system is assumed overtime. There is interesting discussion about it here. The gist is that not every state is equally probable. But given infinite time, there is a chance that it would come back to the original state.

Is Universe finite or infinite?

We don’t really know if the universe is finite or infinite. As per Einstien’s theory of general relativity, universe is finite. It is open but not inifinite. It is like our earth where we keep going in one direction and we will reach that point again coming from another direction.

We never know because the observable universe in which light can reach in our observable time is limited. So we don’t really know whether it is finite or infinite.

We have observed that universe is expanding. The consequence of this is that we will be far from other stars in the galaxy and once our sun is all burnt out, all life on the planet will end.

This is a mind-blowing show to expand your mind. Definitely recommend this.