Jita: Infinite Hotel is a huge hotel in the center of our galaxy. It has an infinite number of rooms, extending to a higher level of time and space through black holes. The room number starts at 1 and goes down indefinitely. One day, all the rooms in this hotel were occupied by guests. At this moment, a UFO pilot came. He is going to another galaxy. Although there was no vacant room, the innkeeper found a room for the driver. He just moved the tenants who used to live in every room to the next room. So the 1 room on the left is vacant for the driver. The next day, five couples came for their honeymoon. Can the Infinite Hotel receive them? That's right. The boss just moved every guest to Room 5, and gave the vacant room 1 to Room 5 to five couples for the weekend. Numerous bubble gum salesmen came to this hotel for meetings. "
Herman: "I can understand how Infinite Hotel receives a limited number of newcomers, but how does it find new rooms for an unlimited number of passengers?"
Keita: "It's simple, my dear Herman. The boss only needs to move the guests in each room to twice the original number of rooms. "
Herman: "That's right! This time, everyone in each room lives in a double room, and there are countless other single rooms, all of which are vacant for bubble gum merchants! "
There are many paradoxes about infinity. The number used for counting is the lowest infinite number in the infinite hierarchy. The number of points in the whole universe is the second infinite number, and the third infinite number is much more than this!
German mathematician George Cantor discovered this level of infinity. He called this new singular energy level Alev Zero, Alef 1, Alef 2 and so on. There are many profound mysteries about Alev numbers, and solving them is one of the most exciting challenges in modern mathematics.
We know that no finite set can establish a one-to-one relationship with its proper subset. This is not true for infinite sets. This seems to violate the ancient law that the whole is greater than the part. In fact, an infinite set can be defined as a set that can correspond to a proper subset.
The owner of Infinite Hotel first proved that the set of all counts (which George Cantor called Alev Zero) can correspond to one of its proper subset, with one element or five elements left. Obviously, this program can be changed to subtract a subset from an Alev zero set, which is also an Alev zero set, and the remaining number will get any finite elements.
There is another way to visualize this subtraction. Imagine that there are two infinitely long measuring sticks on the table, and the zero ends of the two sticks are aimed at the center of the table. Both poles are engraved with lines, which are counted in centimeters. The two rods extend to infinity at the right end, and all the numbers correspond to each other: 0-0, 1- 1, 2-2 and so on. Now imagine moving a stick to the right by n centimeters. After moving, all the numbers on the stick still correspond to the numbers on the fixed stick. If the joystick moves 3 cm, the corresponding teaching on the joystick is 0-3, 1-4, 2-5, ... The moved n cm represents the difference between the lengths of two sticks. However, the length of the two sticks is still zero centimeters of Alev. Since we can make the difference n any value we want, it is obvious that subtracting Alev 0 from Alev 0 is an uncertain operation.
The hotel owner's last strategy is to open an infinite number of rooms. This shows how to subtract Alev zero from Alev zero to get Alev zero. Let each number correspond to each even number one by one, and the rest is Alev zero set composed of all odd numbers.
The set of real numbers forms a high-order infinite set, which Cantor called Alef 1. One of Cantor's brilliant achievements is the famous "diagonal proof", which points out that the elements of Alef 1 cannot form a one-to-one correspondence with the elements of Alef 0. Alef 1 is the number of all points on the line segment. Cantor proved that these points can correspond to points on an infinite line, points on a square and points on an infinite plane. It corresponds to the points in the cube and infinite space, and so on, and corresponds to the points in the hypercube or higher dimensional space. Alef 1 is also called "continuum potential".
Alef 2 is the number of all possible mathematical functions-continuous functions and discontinuous functions. Because any function can be drawn as a curve, we generalize "curve" as including discontinuous curves, so Alef 2 is the number of all possible curves. Similarly, if the curves we refer to are all curves on a stamp, either in an infinite space or in an infinite hyperspace, they are all right, and it is still Alef 2. Cantor also proved that Alef 2 can't correspond to Alef 1.
When an Alev number is raised to its own power, it will produce a higher Alev number, which cannot correspond to the Alev number that produces it one by one. So the ladder of Alev number is infinite.
Is there any extra between Alev numbers? For example, is there a number greater than Alev zero and less than Alef 1? Cantor is convinced that such figures do not exist. His guess became the famous generalized continuum hypothesis.
In 1938, Godel proved that the standard set theory is consistent with the assumption that there is no intermediary overrun. In 1963, paul cohen proved that if people assume that there is an intermediate number, it does not contradict the set theory. In short, the continuum hypothesis is judged by showing that it is "undecidable".
Cohen's research result is that set theory can be divided into Cantor type and non-Cantor type. Cantor set theory assumes that there is no intermediate number between Alev numbers. Non-Cantor set theory assumes that there are infinitely many intermediate numbers. The situation is similar to geometry. After discovering that the hypothesis of parallel lines cannot be proved, it is divided into Euclidean geometry and non-Euclidean geometry.
Students who want to know more about these mysterious surplus phenomena can look at the second chapter "After gugel" of Mathematics and Imagination by Edward Casner and James Newman, and the math game part of Scientific American No.65438+March 0966.
Aleph is the first letter of the Hebrew alphabet.