We have an idea. Pretty naive but working idea to solve the Olbers paradox. The idea is what if our universe is finite? Okay, what if our universe is finite? The problem of Olbers is if the universe is infinite and looks the same everywhere, then looking at whichever angle you will encounter a star, and that star will have roughly speaking the same surface brightness of the sun. And now, what if our universe is finite? In that case, if you look outwards, for example, in this cylinder, there is a probability that you'll encounter a star, and otherwise you don't encounter the star. Let's say, if the lines of the cylinder is the size of the universe, I will specify what do I mean by the size of the universe later. But now, let's just naively assume there is a size of the universe, and then the probability that you encounter a star. So now, if the size is infinite, for sure you will encounter a star. And if the size is finite, we turn this problem into the probability that you can encounter a star. And this probability can be calculated by simply what is the number of density of stars in our universe? And what is the volume of this cylinder from us to the observable universe? Where the radius of the cylinder is defined by the size of the star, which we assume is roughly the size of the sun, all right? Then, it is a simple calculation. So, first of all, we have the number of density of stars, and then the volume of the cylinder, which is the size of our universe in some sense, and then the pi, pi star square. So this is the probability to encounter the star. And actually, this probability is a number that we have previously mentioned. That number is 10 to the -14, which is the ratio between the surface brightness of the night sky divided by the surface brightness of the sun, which is 10 to the -14. This ratio between how darker the night sky is is converted to the problem, what is the probability that we encounter a star in that direction? So this number is 10 to the -14. And then we can simply calculate what is the so called size of the universe from this equation. We solve this Ru, Which is 10 to the -14 divided by n, number density of [stars], and then πr². And what are they? So first, we'll carry on this 10 to the -14, and then the number density of [stars], one can actually count how many [stars] are there. This is a pretty modern calculation. But assume at that time in principle, we can observe the number of [stars] in principle, and the number will be 10 to the 9 per Mpc cube. And Mpc is the union that, nowadays, we usually use in cosmology, Which is somewhere 10 to the 22 metres. And in terms of the light years, is somewhere A million light years, as light travels a million years roughly corresponds to Mpc, all right? And this is the number of density and the radius of the sun square, convert it to Mpc scale, will be 10 to the -26 of, excuse me, this is a -3. This is Mpc square, all right? So, this is the so called size of the universe. And by putting the three numbers together, we'll have the so called size of the universe to be 10 to the 4 Mpc. 10 to the 4 Mpc, which means each Mpc is roughly 10 to the 6 light years. So this is roughly 10 to the 10 light years. This is the so called size of the universe, all right? If we are living in a universe, we are in the center of the universe, and the universe is 10 to the 10 light years large, then we can solve this Olber's paradox. However, here, immediately, you will notice something which is not pleasant that in resolving the Olber's paradox, we have assumed that there is a size of the universe. And if, in some sense, there is the size of the universe, aren't we just living in the center of the universe? And if we live in the center of the universe, that violates the Copernican principle that the universe is not special anywhere. We are living at the centre. This looks not so good, okay? And how do we solve this problem? We remind that light travels at finite speed. So, when we are looking far, when we're looking deep, distant in the universe, we are also looking back in time. Think about the star that we see in our universe. The starlight comes to us and that takes a long time. And that means, by the time we see the star, actually, that was the past of the star, all right? So if we see further and further, that corresponds to a earlier and earlier time of the star and of the corresponding part of the universe, okay? Now, what if our universe instead of having a really finite radius? What if our universe has a finite time from the birth of the universe to now? What if our universe has a finite age? If the universe has a finite age, and this is the beginning of the universe, the birth of the universe. Since we are looking far, then we are looking early, okay? So if this is the birth of the universe, and beyond that, we cannot see the stars. Indeed, there could be space, for example, beyond this circle and then the starlight. Ever since the beginning of the universe traveled to us and it hasn't reached us yet because the age of the universe is finite and the speed of light is finite. So, even if there is space, we are not going to see this star. And now, we arrive at the concept, which is the observable universe. So the universe could be infinite, could be finite. We don't know so far and we may not know what so ever time, but we know there is a size of the observable universe. And the size of the observable universe is the age of the universe multiplying the speed of light. And later, actually, ever tell you, the universe is expanding, so there are some order-one modifications after that. But at this point, let's do the order of magnitude estimates that the size of the observable universe is of order the age of the universe multiplying the speed of light. And we have calculated this number, which is 10 to the 10 light years. So, that corresponds to what's the age of the universe. The age of the universe is then roughly 10 to the 10 years. That means that our universe is not something that existing forever, and our universe is in a state of evolution. And what is the state of the evolution of our universe? What is the equation governing the evolution of the universe? And how matter arises in our universe and how Galaxies arises in our universe, how stars arise in our universe. What is the first ever light, which is emitted it in our universe? And when we are talking about elements, there are a lot of hydrogen in our universe, a lot of helium, but not a lot of other elements in our universe. Why it is the case? And can we understand the ratio between hydrogen and helium in our universe? And after all, finally, why there is a finite age of our universe? What is the infant state of our universe related to the born of the universe? And that will be the part that we are studying in the part of cosmology.