# Exploring the Concept of Infinity: Is There More?

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## Is There Something Bigger than Infinity?

Infinity represents the notion of something that has no end. However, can we truly say that there is a concept larger than infinity?

My young cousin, only two years old, is a delightful little boy. Each time I visit my aunt, I find it hard to leave him behind. He spends his days playing, napping, and watching amusing Cocomelon videos on YouTube. His mother diligently tries to teach him to count to at least twenty, but he can only manage up to seven. It's quite a challenge for him.

For him, counting from zero to ten feels like an insurmountable task. He begins with zero, then counts one, two, three... and after pausing to remember, he continues to four, five, six, and finally seven. Eight, however, seems to be his greatest adversary. I can only imagine that the number eight haunts his dreams.

His struggle is understandable. But you're an adult with a capable mind; surely counting from zero to ten should be a breeze, right?

Well, here's the catch: I didn't specify how to count from zero to ten. If you were asked to count whole numbers, integers, natural numbers, or even odd/even numbers, you would find it straightforward. But if I asked you to count all the real numbers between zero and ten, you would hit a wall. Suddenly, counting becomes impossible!

But don't be disheartened; you're not alone. No one can accomplish this task. But why is that?

The answer is quite simple. You start at zero and then wonder what the next real number after zero could be. Is it 0.001? No, 0.0001 is smaller, and then there's 0.000001, and so on. You find yourself in a mental loop of frustration. The issue is that there is no real number that can directly succeed zero; there's always another number that is smaller.

Now, let's try starting from 2.5. What could be the immediate successor of 2.5? Is it 2.51? No, perhaps it's 2.501, or even 2.5001, or maybe 2.5000000001. The same problem arises again.

You can't even count from zero to one when it comes to real numbers. It's genuinely perplexing.

To understand this better, we need to delve into the concept of infinity.

## What is Infinity?

First, we should clarify what infinity is not. Infinity isn't a number; it simply signifies something that continues endlessly, without limits.

No matter how large a number you can conceive, you cannot replace it with infinity. If a number can be expressed, it isn't infinity. Infinity lies beyond our comprehension.

As a child, I often pondered what the largest number might be. At age five, I believed a thousand was the biggest. As I matured, that number grew to a million, then a billion, and eventually a trillion. But none of these are correct. We can never truly identify the largest number; we can always add one more, making it the new largest number, and this process continues indefinitely.

For example, let's assume X is the largest number in existence. If we add one to it, we get X + 1, which is greater than X. Thus, Y = X + 1 becomes the largest number, and similarly, Y + 1 becomes Z, the new largest number. This cycle goes on endlessly.

So, while we might suggest that infinity is the largest number, that's inaccurate. Infinity is not a number. Instead, we can phrase it this way: the totality of integers, whole numbers, or real numbers is infinite.

## Countable and Uncountable Infinity

You might wonder, since we can't count to infinity, what do we mean by countable and uncountable? This was my initial thought as well. Let's start from the basics with sets.

A set is defined as a collection of distinct elements. These elements can encompass anything: objects, letters, or mathematical constructs such as numbers, points, lines, or even other sets.

This definition indicates that a set can contain either a finite or infinite number of elements.

For instance, if we define A = {1, 5, 6}, this set contains a finite number of elements—specifically, three. Conversely, the set of natural numbers, denoted by ?, comprises all positive integers, which is an infinite set. ? = {1, 2, 3, 4, 5, …}.

Therefore, we categorize sets into two types: finite sets and infinite sets.

### Countable Sets:

A set is considered countable if it has a finite size or if it can be matched with the set of natural numbers, ?.

For example, the set of odd numbers, denoted O = {1, 3, 5…}, is countably infinite because it is a subset of the natural numbers.

Now, let’s discuss the set of rational numbers, ?. This includes all fractions where both the numerator and denominator are integers, such as 1/2, 3/4, 7/2, or 4/3.

? = {M/N | M and N ? ?}

The set of rational numbers is countably infinite because it can be paired with ?. By plotting each positive rational number x/y on a coordinate plane, we can systematically count them while avoiding duplicates since, for instance, 1/1 is the same as 2/2.

This remarkable proof was introduced by Cantor!

### Uncountable Sets:

Now that you grasp the concept of countable sets, let’s explore irrational numbers or, more broadly, all real numbers. The real numbers include decimals, fractions in decimal form, such as 0.5, 0.25, or 5.75, as well as irrational numbers like ? or ?2.

Imagine trying to count all decimal numbers. The issue we previously encountered resurfaces: we can't identify an immediate successor to 0. Is it 0.001? But then 0.00001 is smaller, and 0.000000001 is even smaller. Thus, the range of decimal numbers between any two natural numbers is also infinite. This represents a different kind of infinity: uncountable infinity.

Consequently, the set of real numbers, ?, is uncountable. The reals within the interval (0, 1) are not countable. This is demonstrated through the Cantor diagonalization argument.

Suppose we were to list all decimal expansions of the reals in (0, 1): 0.a1a2a3a4a5… 0.b1b2b3b4b5… 0.c1c2c3c4c5… 0.d1d2d3d4d5… … We can define a decimal x = x1x2x3x4… by selecting x1 ? a1 (or 9), x2 ? b2 (or 9), x3 ? c3 (or 9), and so forth. The decimal expansion of x will differ from the nth element of the list at the nth decimal place. Hence, it represents a number in the interval (0, 1) that is not included in our original counting, confirming that we cannot enumerate the reals in that interval.

Thus, ? is indeed uncountable.

## What Did We Learn?

In summary, there are two types of sets: countable and uncountable. Countable sets can be either finite or infinite, with any infinite set that can be matched with the natural numbers being countable. Conversely, some infinite sets are uncountable, such as the set of all real numbers, ?.

This means that there are two forms of infinity: countable and uncountable. Although the natural numbers are infinite, they are countable. We can enumerate them as 1, 2, 3, 4, 5,… However, the set of real numbers is another story. We cannot even ascertain the immediate successor to 0!

So, while my cousin struggles to count from 0 to 10 due to his young age, you too would find it impossible to enumerate all real numbers between 0 and 10, even in the entire lifespan of the universe. You're constrained by fundamental mathematical principles and logical paradoxes.

A countable set shares the same cardinality as the set of natural numbers, ?. The cardinality of a set quantifies its size, essentially counting the number of elements. An infinite set A is deemed countable if it possesses the same cardinality as ?, indicating a bijection between A and ?.

In conclusion, uncountable infinity surpasses countable infinity!

The notion of uncountable infinity leads to an intriguing thought experiment regarding the devil's number. Consider this scenario:

Suppose you neglected to follow me on Medium and, as a consequence, found yourself condemned to hell for eternity. Fortunately, the devil offers you a chance at redemption by writing a number on a piece of paper, allowing you one guess per day.

Imagine you possess an otherworldly method of tracking all the numbers you've guessed. Will this enable you to escape?

If the devil's number is a whole number, you have a fighting chance. You can start at 0, and eventually, you will guess correctly. The number could be anything—a mundane number like 69 or something far more complex. Regardless, it remains finite, allowing you to eventually reach freedom.

Why does this work? Because the devil's number is a whole number, which has the same cardinality as ?. This principle also applies to sets of rational numbers or even/odd numbers. However, if the devil's number is a real number, you might be out of luck. No matter how many guesses you make, you will perpetually miss countless possibilities, as ? is uncountable.

## Cantor, the Great

Georg Cantor was the pioneer behind the concept of different types of infinities. Sadly, he faced ridicule from his contemporaries and only received recognition posthumously. He also laid the groundwork for set theory, which is integral to modern mathematics.

A vital lesson from his life is that popular opinion doesn't always equate to correctness.

Cantor established foundational principles in mathematics that we continue to follow today. Without his contributions, contemporary mathematics might not have evolved as it has. This article serves as a mere glimpse into one of his theories, but it is insufficient to honor his legacy.

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### Sources:

- Infinite and Infinities
- Uncountable Set
- Types of Numbers