Take a deep breath. There are about 56 sextillion atoms in every breath you take. That's a 5 followed by 22 zeros. There are 7 octillion atoms that make up your body, which is a 7 followed by 27 zeros. The entire planet Earth is composed of 100 quindicillion atoms, or a 1 followed by 50 zeros. In the entire observable universe, which is 46 billion light years across, there are thought to be up to 100,000 quadrillion vigintillion atoms, or a 1 followed by 82 zeros. The biggest number in the cosmos can be arrived at by dividing the total volume of the observable universe by the smallest possible unit of volume, the Planck space. By dividing everything we can see into minuscule cubes, we end up with 1 times 10 to the power of 185 of those cubes. Yet, that is not the biggest number that humans have conceived. Take for example the goliath that is "tree three," as physicist Antonio Padilla at Nottingham University puts it. Everything you think you know is dwarfed into nothing. Welcome to the game of trees.
The Game of Trees
You start with a colored seed, which can give rise to a tree whose branches and seeds can sprout their own trees. The goal is to make as many trees as possible, following two main rules:
The first tree must contain one seed.
The second tree can contain a maximum of two seeds.
The game continues with each level, where the maximum number of seeds allowed corresponds to the level you're at. The game ends and the forest dies if you create a tree that already exists within the forest. Mathematicians have played this game with different numbers of seed colors. Tree one allows only a single seed color, and the forest dies at the first tree. The second seed you add is necessarily a repeat of the first tree, resulting in three trees starting with one color. However, with three colors, tree three is different. While previous forests died quickly, a forest of three colored seeds grows for an extraordinarily long time. Mathematicians have proved that the game must end at some point, but they have not been able to count the number of trees it would take to get there, as it would be too large for the human brain to comprehend.
The Concept of Infinity
When attempting to calculate the length of a square's diagonal, the Pythagoreans realized that not everything could be explained simply using counting numbers. The length of the diagonal was always irrational, never equating to a ratio of two counting numbers. This led to the concept of irrational numbers, which the Pythagoreans deemed evil and to be avoided. However, philosophers over the centuries began contemplating the concept of infinity and the truly limitless. For example, Zeno's paradox states that if you divide a journey into halves an infinite number of times, you will never reach your destination. Some Greeks believed that the concept of infinity belonged to the divine and metaphysical and was not part of the rational world.
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Tried to categorize infinities themselves, there was a difference they claimed between an actual infinity, which is an infinite quantity of things that exist all at once, and a potential infinity, which only comes into existence as a result of a process. So counting to infinity or subdividing a journey an infinite number of times are potential infinities, whereas actual infinities, like an infinitely large object, were still despised and refuted as impossible.
Some 2500 years have since passed since this ancient introspection, and with it has come waves of mathematical and scholarly advancement. No longer tied to cults or secret societies, numbers could be analyzed for what they really were, rather than what they signified. And the same was true for our ideas of infinity.
Marked by a huge leap forward in the late 19th century by a German mathematician named Georg Cantor, Cantor was the first to consider that there truly are different kinds of infinity. He described how there are an infinite number of counting numbers, but also an infinite number of even numbers too. Both groups are infinite in size, despite the fact that the list of counting numbers completely contains the list of even numbers plus the odd numbers as well. This apparent paradox is possible purely because of the bizarre nature of infinity.
Each of the even numbers can be put in a numbered list, paired with a corresponding counting number. So 2 is number one, four is number two, six is number three, and so on, right up to infinity. You'll always be able to match an even number to a counting number, so they are both the same size. But Cantor also showed that there are actually infinities of different sizes when you consider subsets within an infinite number of counting numbers.
Take the four members of the Beatles: John, Paul, George, and Ringo. How many different combinations or subsets of members could you form from among this group? You could have each of the four by themselves, pairs like John and Ringo or Paul and George, trios like John, Paul, and Ringo, and you can have a set of zero, no Beatles at all. In total, there are 16 possible configurations of the four Beatles, many more than the original number. The same applies to infinity because of this multiplying nature of subsets. There will always be more subsets than there are counting numbers. There are indeed infinite sets of infinite counting numbers, but one infinity is necessarily larger than the other.
If you find this to be twisting your brain to breaking point, you're not alone. Georg Cantor himself suffered several nervous breakdowns over the course of his career and spent the last year of his life in a mental hospital. Nevertheless, we must continue to consider the infinite if we hope to understand the past, present, and future of the cosmos. For all their complexity, Cantor's transfinite mathematics and the Greeks' desperate reasoning with impossible paradoxes were limited to the contemplation of numbers alone. The Greeks desperately tried to avoid any contact between the infinite and reality.
But astronomers now find themselves confronted with actual infinities, from the infinite densities at the heart of supermassive black holes to the potentially infinite expanse of the universe as a whole. Scientists are now equipped with the mathematical and practical tools to tackle these problems head-on and face the many oddities of an infinite cosmos and what they would mean for our tiny, fleeting existence.
As one sets foot on North Sentinel Island, the familiarity of the terrain is overwhelming. Each tree, bush, and rock is as recognizable as the faces of one's children. Gradually, the trees thin out and reveal a sandy beach, beyond which lies a vast expanse of blue. This is the man's entire universe. The island is his world, and everything he can see is all there is.
But one afternoon, something new arrives in the man's universe. A white speck at first, too small to make out at the edge of the ocean, it gradually grows in size as it approaches the island. The young man and his companions are terrified of these new alien beings that have descended silently into their world. They throw rocks and sticks at the gigantic vessel before retreating back under the cover of trees.
If the island and its surrounding ocean were all there was, then where did these aliens and their giant vessel come from? If the man ventured back out from the safety of the trees, he would have seen the white boat shrink and then disappear beyond the horizon of his own universe.
