Buzz
In the late 1800s, German mathematician Georg Cantor unveiled the concept of 'transfinite' mathematics, diving into the realm beyond infinity. His pioneering work revealed a universe where numbers transcend infinity, and equations defy the conventional rules of arithmetic. Needless to say, this isn’t the kind of math you encountered in high school.
Cantor’s discoveries were met with intense opposition and scorn from some of the most prominent mathematical minds of his time. Yet, over time, his ideas gained widespread acceptance and formed the foundation for set theory—a key pillar that supports much of modern mathematics.
10. Infinity Plus One (Or Two, Or Infinity) Still Equals Infinity
It turns out that this childhood concept holds more truth than we might have imagined. Due to the very nature of infinity, adding, subtracting, multiplying, or dividing any number by infinity still results in infinity. This concept is illustrated in the famous thought experiment known as Hilbert’s hotel paradox:
Imagine a hotel with an infinite number of rooms. A tired traveler arrives and asks for a room, but is told that all rooms are currently occupied. How is it possible that there are no available rooms when the hotel has an infinite capacity? What solution can the traveler come up with?
The solution lies in the traveler asking the occupant of room one to move to room two, the person in room two to shift to room three, and so on. This chain continues, making room one available for the traveler. Infinity is incredibly flexible, capable of expanding or contracting as needed, whether accommodating a single traveler or an overwhelming number, like a googolplex (a real number!).
9. There are as many odd numbers as there are numbers ending in 123 or 423.
Infinity’s flexibility arises from the concept of a 'one-to-one correspondence.' Simply put, you can match any whole number (0, 1, 2, 3, 4...) with a positive even number (0, 2, 4, 6, 8...). For example, zero pairs with zero, one pairs with two, two pairs with four, and so on.
Because the two sets of numbers can be paired in this way, we can confidently say they are of equal size. This thought experiment, known as the Galileo Paradox after its renowned creator, illustrates that the size of infinity cannot be manipulated using basic arithmetic tools like division or simple addition of finite numbers. To truly understand infinity, a more advanced approach is needed.
8. Some infinities are larger than others.
The flip side of the one-to-one correspondence principle is that if there exists an infinite series of numbers that still has numbers left after being paired with another infinite series, we can say that the first series is actually larger than the second. It may seem counterintuitive, but consider this example: the infinite set of whole numbers (0, 1, 2, 3...) is smaller than the infinite set of irrational numbers. Recall from high school math that irrational numbers like pi have decimal expansions that go on forever (3.1415...). Cantor proved, using a clever and relatively simple method, that the infinite set of irrational numbers is actually larger than the set of whole numbers.
Cantor started by assuming that irrational numbers could be paired with whole numbers and wrote down a list of numbers between zero and one. (Note: These are my own random numbers, but the point remains the same.) There are infinite such sequences:
0.1435... paired with 0 0.7683... paired with 1 0.1982... paired with 2 0.9837... paired with 3
And so on. Now, by taking the first digit from the first number, the second digit from the second number, and continuing this pattern, you can create a new number from this series. For the above numbers, this would be 0.1687...
In a collection of numbers, you might come across a value like 0.1687... somewhere within. However, if you increase each digit by one, the result becomes 0.2798..., which is guaranteed to be absent from the original set, as it differs by at least one digit. This demonstrates that even after attempting to match irrational numbers with whole numbers, there remain irrational numbers that can't be matched. This allows us to conclude that the set of irrational numbers is indeed larger than the set of whole numbers.
If you find that unbelievable, just wait until you see what comes next...
7. The Infinite Hierarchy of Infinities
Cantor also demonstrated that, much like the infinite set of whole numbers has a distinct level of infinity compared to irrational numbers, there exists an even greater infinity, one larger than the set of irrational numbers. Moreover, any new infinity added to a larger infinity will still be classified as the larger infinity, similar to how infinity plus one still equals infinity.
To put it simply, consider an endless sequence of numbers (such as 0, 1, 2, 3, ...) and imagine creating a larger infinite set by combining all possible groupings from this original sequence. In mathematics, this is referred to as a power set. For whole numbers, the power set includes not only 1, 2, 3... but also every possible combination, such as 1 billion and 1, 2, 13, 2 million, etc. Once you form your initial power set, there's nothing stopping you from generating a power set of the power set, or even a power set of a power set of a power set, continuing indefinitely.
6. The Mental Toll of Georg Cantor’s Groundbreaking Work Led to His Breakdown
As you can imagine, obsessing over such complex concepts for too long can take a toll on one's grasp of reality, and that's precisely what happened to Cantor. He postulated that the next stage of infinity beyond whole numbers was the realm of irrational numbers; however, the problem was that he couldn’t substantiate this theory.
This well-known mathematical dilemma, known as the continuum hypothesis, led Cantor to declare that God himself had revealed its truth to him. The harsh criticisms of his work further contributed to his mental decline, ultimately causing him to spend his remaining years alternating between hospitals while also attempting to prove that Francis Bacon was the true author of Shakespeare's plays.
5. The Unsolvable Problem That Drove Cantor to Madness
Some individuals have attempted to establish a solid foundation for mathematics by relying on a series of axioms, which are propositions believed to be so self-evident that they don't require further justification. (For example: One cannot equal two. Why? Because!)
In the 1960s, mathematician Paul Cohen demonstrated that the continuum hypothesis cannot be solved if we accept the most widely used axioms as true. Nevertheless, mathematical work continues to proceed under the assumption that the axioms are valid and that the continuum hypothesis is false, as well as the opposite assumption that both the axioms and the continuum hypothesis are true. Mathematicians regard these conflicting assumptions about the continuum hypothesis as part of distinct 'mathematical universes,' as we cannot definitively prove the truth of either one.
4. The Infinity Symbol Selected by Cantor is a Hebrew Letter
Just as astronomers and biologists have naming rights for new discoveries, mathematicians who introduce new concepts or important values also have a say in what they are called. You'd expect to see more unconventional symbols like Klingon characters in higher-level mathematics, but no—mathematicians tend to stick to traditional Greek letters. This reliance on Greek symbols leads to multiple meanings for various letters, depending on the area of math, because we simply have more constants and concepts than Greek letters can cover.
While Cantor's religious beliefs remain a subject of historical debate, he perceived his work as a way to approach the divine through mathematics. For this reason, he chose to represent the different levels of infinity with the first letter of the Hebrew alphabet: aleph. The set of all whole numbers was labeled aleph-naught, or aleph with a zero subscript. The next level of infinity would be aleph-one, which, as previously mentioned, may or may not represent the number of irrational numbers.
3. Real-World Applications for This Concept
Much like other areas of mathematics, concepts that began as abstract theoretical exercises have been found to have significant implications in the physical sciences. For instance, certain equations in quantum mechanics result in infinity. While physicists adjust these equations to make calculations feasible, it's still unclear whether such adjustments are justifiable, considering what we know about transfinite mathematics.
In the field of cosmology, questions such as whether the universe is infinite in size, whether space can be divided infinitely, whether the universe will perpetually expand, or whether multiple infinite universes exist, all rely on principles of infinite logic. Some scientists have even explored the use of Hilbert’s hotel paradox in both quantum and classical optics.
2. Infinity Minus Infinity Is Not Equal to Zero
Infinity minus infinity is undefined, much like division by zero is undefined.
To illustrate why this happens, consider that infinity plus one equals infinity ([infinity + 1] = [infinity]). If we subtract infinity from both sides, we're left with 1 = 0. In a similar vein, and for many of the same reasons, infinity divided by infinity does not equal one, but rather is undefined.
1. There's a Level of Infinity Where Infinity Plus One Isn't Equal to One Plus Infinity
Along with the aleph numbers, Cantor also introduced omega numbers. The first omega number is defined as the smallest number greater than the total number of whole numbers, or the first number following aleph-naught. To illustrate with Hilbert’s hotel analogy, if the number of rooms is aleph-naught, then the first omega number is a shack outside the hotel. The next omega number is simply omega plus one. What this implies is that one plus omega is distinct from omega plus one, because the former simply absorbs the one (since infinity is malleable), whereas the one after omega signifies the next level.
Unfortunately, grasping a more technical proof of this is beyond the intellectual capacity of your humble author, but I read it in a book, so it must be true.
