Are there levels of infinity

The natural numbers are a part of the integers. The next thing we can try is to include all the numbers that can be written as fractions, creating the set of rational numbers GLOSSARY rational numbers The set of all numbers, both positive and negative, that can be written as a fraction.

Equivalently, the set of all numbers whose decimal expansions either terminate or eventually repeat. The natural numbers and the integers are a part of the rational numbers. Difficult … but not impossible. Cantor showed how this could be done by thinking about things a little differently. The key is to realise that in order to pair off the rational numbers with the natural numbers, we just need to find a way to list the rational numbers in some sort of order.

We can then pair the first rational number in our list with the natural number 1, the second rational in the list with 2, and so on. On the surface, this seems a tall order—how can we can list the rational numbers, which include fractions, in an order that captures every single one?

But by doing a few mental zigzags, it turns out that it is possible to make such a list. To start, we notice that every fraction is made from a pair of natural numbers—one on the top of the fraction the numerator and one on the bottom the denominator. There are two caveats. First, we cannot have 0 on the denominator of the fraction. We learn early in school that there are four basic operations in mathematics: addition, subtraction, multiplication and division.

To mathematicians, however, there are only two basic operations: addition and multiplication. We get subtraction and division for free, as the opposites of addition and multiplication. More specifically, every subtraction question can be thought of as an addition question, and every division question can be thought of as a multiplication question.

For instance, a typical multiplication question might look like:. But what does this mean? In other words,. So a division problem is really a multiplication problem with the unknown number on the other side of the equation.

But there is no number that works! So, take the first digit of the first number, the second digit of the second number, the third digit of the third number, and so on:.

From our first real number we get a 5, our second number a 3, and our third number a 1. We make a new number by taking each of these digits, and adding 1 to them flipping around to a 0 if my original digit is 9 , giving us the number 0.

This new "diagonal" number is definitely a real number — it has a decimal expansion. But it is different from all the numbers on the list: its first digit is different from the first digit of our first number, its second digit is different from the second digit of our second number, and so on. We have made a new real number that does not show up on our list.

This contradicts our main assumption that every real number appears somewhere in the correspondence. We mentioned before that the details of the correspondence did not matter. This is because, no matter what alignment we try between the real numbers and the natural numbers, we can do the same diagonal trick above, making a number that does not show up in the correspondence. This shows that the reals are not countably infinite.

No matter what we try, there is no way to make a one to one pairing up of the natural numbers and the real numbers. These two sets are not the same size. This leads to the profound and somewhat uncomfortable realization that there must be multiple levels of infinity — the natural numbers and the real numbers are both infinite sets, but the reals form a set that is vastly larger than the naturals — they represent some "higher level" of infinity.

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It often indicates a user profile. Log out. US Markets Loading H M S In the news. Andy Kiersz. Jerome Keisler, emeritus professor of mathematics at the University of Wisconsin, Madison.

He proposed a technique for measuring complexity and managed to prove that mathematical theories can be sorted into at least two classes: those that are minimally complex and those that are maximally complex. Much work in the field is motivated in part by a desire to understand that question. Keisler describes complexity as the range of things that can happen in a theory — and theories where more things can happen are more complex than theories where fewer things can happen.

A little more than a decade after Keisler introduced his order, Shelah published an influential book, which included an important chapter showing that there are naturally occurring jumps in complexity — dividing lines that distinguish more complex theories from less complex ones. In , she and Shelah started working together to better understand the structure of the order. Malliaris and Shelah eyed two properties in particular.

They already knew that the first one causes maximal complexity. They wanted to know whether the second one did as well. As their work progressed, they realized that this question was parallel to the question of whether p and t are equal. In , Malliaris and Shelah published a page paper that solved both problems: They proved that the two properties are equally complex they both cause maximal complexity , and they proved that p equals t.

This past July, Malliaris and Shelah were awarded the Hausdorff medal, one of the top prizes in set theory. The honor reflects the surprising, and surprisingly powerful, nature of their proof. Most mathematicians had expected that p was less than t , and that a proof of that inequality would be impossible within the framework of set theory.

Malliaris and Shelah proved that the two infinities are equal. Their work also revealed that the relationship between p and t has much more depth to it than mathematicians had realized. Instead, Malliaris and Shelah proved that p and t are equal by cutting a path between model theory and set theory that is already opening new frontiers of research in both fields.

Their work also finally puts to rest a problem that mathematicians had hoped would help settle the continuum hypothesis. Clarification: On September 12, this article was revised to clarify that mathematicians in the first half of the 20th century wondered if the continuum hypothesis was true. As the article states, the question was largely put to rest with the work of Paul Cohen.

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