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10 facts from the bizarre world of infinite mathematics



In the late 19th century, the German mathematician Georg Cantor discovered "transfinite" mathematics or mathematics beyond infinity. With this early work we have been introduced to a world where there are numbers greater than infinite and equations that do not follow the arithmetical rules of common sense. Suffice to say, that's probably not what you learned in high school.

Cantor's work was initially controversial and vitriolic was attacked by some of the most important mathematical figures of his time. However, it has gradually been accepted as a canon and has helped pave the way for set theory, which itself poses a potential underweight to all mathematics.

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Infinity plus one (or two or infinity) equals infinity


It turns out that this old adage of childhood has something about it. Given the nature of infinity, any number that is added, subtracted, or shared by it is infinite. This can be seen in a classic infinity puzzle known as Hilbert's Hotel Paradox:

There is a hotel with an infinite number of rooms. A tired guest arrives and asks for a room, but is informed that all rooms are occupied. How can the hotel have no more rooms, as it has an infinite number of rooms? What should the traveler do?

The answer is that the traveler should ask that the person in room one go in room two, the person in room two in room three, and so on. , , and she takes first place. Infinity is infinitely elastic and can be arbitrarily expanded or shrunk, no matter what it takes, whether it's a traveler or a Googolplex traveler (yes, that's an actual number). [1]

9 There Are So Many Odd Numbers (And So Many Numbers Ending in 123 or 423) How Numbers Are


Infinity is so malleable because, like Hilbert's Hotel, every row of infinite numbers in a so-called "one-to-one correspondence" can be inserted. with some infinite part of this series. In layman terms this means that if you take all positive integers (0, 1, 2, 3, 4 …) and all positive even numbers (0, 2, 4, 6, 8 …), everyone of the integers can be matched with an even number. So zero can be compared to zero, you can agree with two, two with four and so on.

Since the two series (or "sets") of the numbers match for each number, we can rightly say that they are the same size. This thought experiment, called the Galileo paradox after its famous discoverer, shows that the magnitude of infinity can not be changed by the crude tools of arithmetical principle division or the addition of finite numbers. For that you need something more advanced. [2]

8 Some infinities are greater than others


The flip side of the one-to-one correspondence is, if any, an infinite series of numbers that still exist Having numbers after being matched with another infinite series, we can say that the former sequence of infinities is actually greater than the infinity with which it was brought together. This may seem impossible, but you can probably intuitively grasp a case in which this is true: the infinite number of integers (0, 1, 2, 3, …) is smaller than the infinite number of irrational numbers. If you remember high school math, irrational numbers are numbers like pi, which have a series of decimal numbers that go on forever (3.1415 …). Cantor showed that the infinite number of irrational numbers is greater than the infinite number of integers, using an ingenious but simple trick (relative to most groundbreaking mathematical proofs). [3]

He assumed that irrational numbers could be associated with integers and write down a series of numbers between zero and one. (Okay, these are my own random numbers from mashing the keyboard, but you get the point.) There are an infinite number of these lines:

0.1435. , , paired with 0
0.7683. , , paired with 1
0.1982. , , paired with 2
0,9837. , , paired with 3

And so on. You can then create a number from this series by selecting the first digit in the first line, the second digit in the second line, and so on. For the above numbers this would be 0.1687. ,

Now there could be a number of 0.1687. , , somewhere in this stack of numbers. However, if you add a value to each digit, the number becomes 0.2798. , , , and this number can not be in the stack because it is by definition different from each of the numbers in the stack by at least one digit. Therefore, there are still irrational numbers left after trying to match them to normal integers. Therefore, we can say that the infinite number of irrational numbers is greater than the infinite number of integers.

If you think that's crazy, stick to your hat. ,

7 There are infinitely many levels of infinites


Cantor also showed that just like the number of infinite integers, there is a very different level of infinity than the number of irrational numbers, also a type of infinity greater than the number of irrational numbers, one level of infinity above, another above it, and so on, through (you guessed it) infinity. Moreover, each infinity plane added to a higher infinity level automatically adds to the higher infinity plane in the same way as infinity plus one infinity. [4]

The Reader's Digest version of why this is the case that you have an infinite series of numbers (for example 0, 1, 2, 3 …) Can take and then make a bigger infinite series by taking the number of all different possible combinations of the numbers in the original series. In mathematics, this is called a power set. So for the integers, the power set would not just include 1, 2, 3. , , but also any combination of numbers in this infinite series of numbers including 1 billion and 1, 2, 13, 2 million. , , etc. Once you have made your first power connection, there is no reason why you can not generate power from the generator set or a generator set of a generator set generator. , ,

6 All of this eventually drove George Cantor crazy

As you can imagine, all of this can do too much in your sense of reality, and that's exactly what happened to the discoverer. Cantor believed that the "next" level of infinity after the integers was the number of irrational numbers; The only problem was that he could not prove it.

This famous mathematical problem, called the continuum hypothesis (he finally started saying that God revealed to him that the continuum hypothesis is true), combined with the malicious attacks on his work, eventually led to a psychological collapse and collapse he spent the rest of his days in and out of hospitals while trying to prove that Francis Bacon had written Shakespeare's plays [5]

5 The problem that drove Cantor mad is unsolvable


Some people have tried to establish a rigorous foundation for mathematics by using a set of axioms or statements that are said to be so mean that they can be trusted without prior explanation. (For example, you can not be two, why? Because!)

In the 1960s, mathematician Paul Cohen proved that the continuum hypothesis is not solvable if we assume that the most commonly used axioms are true. To this day, however, mathematical work continues to be conducted on the assumption that the axioms are true and that the continuum hypothesis is false, as well as the inverse assumption that the conventional axioms are true, as well as the continuum hypothesis. Mathematicians consider the various assumptions about the continuum hypothesis as belonging to different "mathematical universes" because we can not prove that one or the other is true. [6]

4 The Symbol of Infinity This Cantor Selected a Hebrew Letter


Mathematicians who discover a new concept or value, such as astronomers and biologists, at least get some information about what the name is. Given this kind of power, you would think that today there would be more Klingon characters in high-level math, but no. As creative as mathematicians are, hardly any of them want to deviate from the very conventional Greek symbols, which is why different Greek letters can mean so many different things, depending on which branch of mathematics you use – we just have so many mathematical constants and concepts Greek letters.

While his religious background is still discussed by historians, Cantor saw what he was doing to approach the divine through mathematics, so he decided that the various levels of infinity would be symbolized by the first letter in the Hebrew alphabet : Aleph. The set of integers would be aleph-noaught or aleph with a zero index. The next highest infinity would be aleph-one, which, as we have mentioned, may or may not be the number of irrational numbers. [7]

3 There's a Plane of Infinity Where Infinity Plus One plus Plus Infinity does not equal


In addition to the Aleph numbers, Cantor has also found omega numbers. The first omega number is defined as the smallest number that is greater than the number of integers, or the first number after the aleph zero. To refer back to Hilbert's hotel example, if the number of rooms is alephonless, then the first omega number is a cabin outside the hotel. The next omega number after that is just Omega plus one. This means, however, that one plus omega is different from omega plus one, since one in the former would simply be absorbed by omega (since infinity is malleable), while the one after omega represents the next step. [8]

Unfortunately, understanding a technical proof replaces the abilities of the mind of your humble author, but I read it in a book, so it must be true.

2 Infinity Minus Infinity is not equal to zero


Infinity minus infinity is undefined in the same way as the division by zero is undefined.

To give an example of why this is so, since infinity plus one is infinite ([infinity + 1] = [infinity]), if we subtract the infinity from both sides, we have 1 = 0. Similar and for many of For the same reasons, infinity divided by infinity is not one, but also undefined. [9]

1 That Has Real Scientific Applications


Like many other areas of mathematics, What began as a purely theoretical thought experiment has implications for the hard sciences. For example, some quantum mechanical equations add up to infinity; In practice, physicists optimize the equation to make computations feasible, but it is not clear if this is justified knowing what we know about transfinite mathematics.

In cosmology, whether the universe is infinitely large, space is infinitely divisible The universe will expand forever, or if there are infinite universes, all are open questions based on infinite logic. [10] Some researchers even found Hilbert's Hotel Paradox in both quantum optics and classic optics.

Stephen Cranney is a postdoctoral fellow in sociology and demography at Baylor University, specializing in fertility, religion and sexuality. He has recently freed himself from his US public school and now values ​​mathematics as the art form that it is.



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