How Big Is Infinity?

A hologram of Tony Stark says "I love you 3000" at the end of the movie. The touching moment is similar to an earlier scene in which the two are trying to quantify their love for each other. Robert Downey Jr., who plays Stark, said the line was inspired by his own children.

The game can be enjoyable.

I adore you 10.

I adore you 100.

I adore you 101.

This is how I came to know the word "ooochplex" in my home. We all know where this argument ends.

I adore you!

Is that true? I love you so much.

Patrick Honner is a high school teacher from New York.

You can see all the academy columns.

Children develop a fascination with the concept of infinite when they encounter it on the playground or at sleep time. Some of those children grow up to be mathematicians, and some of them are discovering new and surprising facts about the world around them.

Did you know that some infinities are larger than others? We don't know if there are other infinities between the two we know best. The second question has been pondered by mathematicians for at least a century and recent work has changed the way people think about it.

To answer questions about the size of infinite sets, we should start with sets that are easy to count. A finite set is a collection of objects that are finite in number.

It's easy to determine the size of a finite set by counting the elements. When the set is finite, you know when to stop counting and when you know the size of the set.

The strategy doesn't work with infinite sets The set of numbers is called N. Zero is not a natural number, but that doesn't affect our investigations.

The mathbbN is 0,1,2,3,4,5.

The size of this set is not known. Trying to count the number of elements won't work since there isn't a big natural number. One solution is to simply declare the size of the infinite set to be infinite, which isn't wrong, but when you explore other infinite sets, you realize it isn't quite right.

A set of real numbers, which are all the numbers expressible in a decimal expansion, like 7, 3.2, 8.015, or an infinite expansion like sqrt2, can be considered. The set of reals must be as large as the set of natural numbers since every natural number is a real number.

There is something odd about declaring the size of the set of real numbers to be the same size as the natural numbers. Pick any two numbers and see why. There are finitely many natural numbers between those two numbers. There will always be many real numbers between them.

There will always be infinitely many real numbers in between even if the two numbers are close. It doesn't mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two sets.

In the late 19th century, the mathematician was investigating this. He showed that the two sets are different in size. We need to understand how to compare infinite sets to understand how he did that. Functions are a staple of math class.

Functions can be thought of in many different ways, but here we will think of a function.

The set of natural numbers is called N. All of the natural numbers will be taken for the other set. Our sets are here.

mathbbN is 0,1,2,3,4,... S is 0,2,4,6,8.

The elements of N are turned into the elements of S using a simple function. If we think of the elements of N as the inputs of f(x), the outputs will be elements of S.

The function f turns inputs from N into outputs in S can be visualized by lining up the elements of the two sets side by side and using arrows to show how the function turns inputs from N into outputs in S.

Each element of N is assigned one element of S. Functions do what they do, but f(x) does something different. Everything is assigned to something in N first. Every element of S is the image of an element of N under the function f The even number 3,472 is in S, and we can find an x in N such that it is 3,472. It is said that the function f(x) maps N onto S. Nothing in S is missed as the function f(x) turns inputs from N into outputs in S.

If two numbers are different, then their doubles are different, and their outputs in S are also different. We say that f(x) is 1-to-1 and that it's anjective. The key is that there is only one element in N for every element in S.

The features of f(x) combine well. The function f(x) creates a perfect match between the elements of N and S. The function f(x) pairs the elements of N and S.

A function that is injective and surjective is called a bijection, and it creates a 1- to-1 correspondence between the two sets. One way to show that two infinite sets have the same size is by showing that every element in one set has a partner in the other.

The two infinite sets N and S are the same size as our function f(x). N is a natural number, so it has everything in S and more. N should be bigger than S. Yes, if we were dealing with finite sets. It is possible for one infinite set to contain another and still be the same size. One of the surprising properties of infinite sets is this.

It is possible that there are infinite sets of different sizes. We looked at the different natures of the infinite sets of real and natural numbers and found that they have different sizes. He used his famous diagonal argument to do so.

Let's focus on the infinitely many real numbers between zero and 1 since there are infinitely many real numbers. There is a possibility that each of these numbers can be thought of as a decimal expansion.

We don't include the number zero in our set because not all the digits are zero

The diagonal argument begins with a question about what would happen if a bijection existed between the numbers. You could use the function to match up the real numbers between zero and 1 with a natural number if the function existed. An ordered list of the matchings would be similar to this.

The genius of the diagonal argument is that you can use this list to make a real number that can't be on the list. If you want to build a real number digit by digit, start by making the first, second, and third digits different from each other.

The diagonal of the list defines the real number. Are it on the list? It is not possible to be the first number on the list. It can't be the second number as it has a different second digit. It can't be the nth number because it has a different digit. This new number, which is between zero and 1, can't be on the list because it's true for everyone.

The real numbers were supposed to be on the list. There is an assumption that there isn't a bijection between the numbers and the reals. These infinite sets are different in size. A little more work with functions can show that the set of all real numbers is the same size as the set of all the reals between zero and 1.

The termcardinality is used for the size of an infinite set. The diagonal argument shows that the reals are better than the numbers. The name of the number is aleph_0. This is the smallest infinite cardinal.

For more than a century, mathematicians have been confused by a simple question: Is aleph_1 the true number? Are there any other infinities between the numbers? He wasn't able to prove that the answer was no, but he did think it was. The continuum hypothesis was the number one hypothesis when David Hilbert put together his famous list of open problems in mathematics.

Progress has led to new mysteries after a century. Kurt Gdel proved in 1940 that it is not possible to prove that there is an infinite number between the numbers and the reals. Two decades later, the mathematician Paul Cohen proved that it is not possible to prove that an infinite doesn't exist. The continuum hypothesis can't be proved either way.

The results established the independence of the hypothesis. The commonly accepted rules of sets don't say enough about whether or not there is an infinite number between the numbers and the reals. It has led mathematicians in a different direction. New fundamental rules for infinite sets are needed to help fill in the gaps and explain what's already known.

The next generation of mathematicians might benefit from saying "My love for you is independent of the axioms."

Exercises

This is the first thing. The set of positive odd natural numbers are called T's. Is T larger than N, or the same size as T?

There are two There is a correspondence between the set of natural numbers, N, and the set of integers, mathbbZ.

There are three. The set of real numbers between zero and 1 and the set of real numbers greater than zero are the same.

There are four. There is a function between the set of real numbers between zero and 1 and the set of all real numbers

Click for the first answer.

The same size. You can use the function f(x) = 2x+1 to turn inputs from ℕ into outputs in T, and this does so in a way that is both surjective (onto) and injective (1-1). This function is a bijection between ℕ and T, and since a bijection exists, the sets have the same size.

You can click for answer 2.

One way is to visualize the list of matching pairs, like this:

You can also try to define a function that matches up the elements. This function,

If $n$ is even, f(n) is the beginning and end of the case.

There is a map of N on mathbbZ. There are as many different numbers as there are different types of numbers.

Click for the third answer.

There are many possibilities, but the simplest is f(x) There is an image of a real number between zero and 1. To find which number is coupled with, say, 102, set the number to fracx1-x and solve for it.

fracx1-x

There is a correlation between the number of x and the number of 1x.

There is a correlation between the number 102 and the number 103.

The x we found is between zero and 1 For any number, like 102, we can find an input that gets mapped onto it. One way to see that f(x) is injective is to graph it and observe that every horizontal line passes through the graph of f(x) at least once.

Click for the correct answer.

As with exercise 3, there are multiple functions that can work, but a standard approach is to use a transformation of the tangent function. For the domain -frac{π}{2}, the standard tangent function, tan(x) , is 1-1 and maps -frac{π}{2} onto the set of all real numbers.

You can alter the domain of this function with a transformation. For example, we can shrink the domain from -frac{π}{2} < x to -frac{1}{2} by multiplying the input by π. In other words, the function tan(πx) maps -frac{1}{2} onto the set of all real numbers. We can then shift this domain over using a translation, ending up with the function f(x) = tan(π(x-frac{1}{2})). This function is 1-1 and maps the real numbers 0 onto the set of all real numbers. This bijection proves that there are as many real numbers between zero and 1 as there are real numbers.