First, there was Infinity, Second, there was Infinity…
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Albert Einstein once said, “Two things are infinite: the universe and human stupidity; and I’m not sure about the universe.” This leads me to wonder many things: One, will we ever evolve into intelligent beings? Two, is the universe really infinite? And three, what is infinity? For the sake of brevity, I am going to address the third question.
Well, infinity has been around since forever. Pun intended.
But, let’s start with the basic definition. Infinity is not a number. So, if you were thinking I was going to have a podcast where I count to infinity, that’s not going to happen. It could be fun, but then maybe not.
Infinity is a concept. It’s this idea that infinity is larger than any number you can think of. It’s also smaller than any number that you can think of. In both cases, they that go on forever and ever and ever without end.
In other words, they are unbounded.
In a small nutshell, a cardinal number is a number where one stops counting. Infinity with Cardinals is Infinity with numbers. We count up: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The thing is that we can’t actually count to infinity. As a result, it’s a concept, much like its smallest infinity, which is also known as an Aleph-null. Aleph-null is the very first cardinal number in a series of cardinal numbers.
Then we have infinity with ordinals. Ordinals allow us to arrange a collection of sets. So, I could have a set of three shoes, a set of seven socks, and a set of one pair of underwear. If I wanted to put them in order, hence the term ordinal, I would say:
First, I have a pair of underwear,
Second, I have three shoes,
Third, I have seven socks.
What I love about this set is that a pair of underwear is really only one underwear, three shoes mean you are missing a shoe, and seven socks mean that you only have clean socks for 3 days and the third sock sits in the drawer for an infinite amount of time.
The thing about Ordinals is that they have a wide variety of different sets, and some of these Ordinals hold sets with bigger countable numbers, like infinity to the power of infinity. Once they reach that size, they are uncountable. As a result, Ordinals are bigger than Cardinals are.
Other types of infinite numbers can be used in various areas of mathematics. These include Beth numbers, Superreals, which are used in Field Theory, Surreals, which are used in Game Theory, Super Complex, and others. In Calculus, we work with infinity in such a way that we take things to the limit.
Infinity is a concept that goes back to as early as 450 BCE when Zeno of Elea presented his paradox that includes repeated division by two. It means that if you divide a whole into a half and then divide that half into another half, and then divide that half into another half, and then divide that half into yet another half, you have infinite halves. Sometimes I feel this way about a goal. When I want to accomplish something, it feels like the distance between where I’m at and where I want to be is just a never-ending path of infinite halves. I’m sure some of you can relate.
Then in 350 BCE, during Aristotle’s time, Aristotle believed that there were aspects of the world that utilized the concept of Apeiron. Apeiron is a Greek word that means unbounded like infinity. The Greeks considered apeiron as a bad word as it alluded to the concept of chaos: a crumbled piece of paper was an apeiron, jagged edges described an apeiron. But out of chaos came order. Thus, Aristotle believed that the set of natural numbers could be potentially infinite.
After Aristotle, this concept Infinity was used sporadically. Then in the third century, a Neoplatonic philosopher by the name of Plotinus developed a philosophy that presented the concept of an infinite God (also known as The One) who could think infinite thoughts. What is interesting about this is that Plotinus was a Neoplatonic philosopher whose theories were foundational to academic paganism, which was a philosophy that Theon and his daughter, Hypatia, taught. Paganism was considered repugnant and disgusting to the church. However, 200 years later, Augustine, who eventually was canonized as a saint, incorporated the philosophy of Plotinus into his teachings. Thus, Augustine preached that God was infinite, and God could think infinite thoughts.
So now, we enter into the Medieval Age, where this concept of a number is part of spiritual teachings. It wasn’t until around the time of 1200, that infinity became a mathematical concept. Enter Fibonacci, AKA Leonardo of Pisa. Fibonacci proposed a sequence of numbers created by adding the two numbers before it.
The sequence starts, 0, 1, 1, 2, 3, 5. The next number is 8, since 5+3=8. The next number is 13, since 8+5=13. So the sequence goes 0 1 1 2 3 5 8 13, etc., etc., etc., all the way up to infinity. What is cool about the Fibonacci sequence that it can be visualized. When you take these values and make them the length and width of a square, position the squares next to each other, and then draw a curve, you get the Fibonacci spiral. This spiral is a mathematical representation of nature. You can see the Fibonacci spiral in plants, in our galaxies, in the swirl of water, it is everywhere, and it is beautiful.
We first start with two squares, each with sides equal to 1, thus giving an area of 1.
1 x 1 = 1
Next to those two squares, we have a square with side lengths of 2. This square has an area of 4.
2 x 2 = 4
Next to those three squares, we have a square with side lengths of 3, thus with an area of 9.
3 x 3 = 9
Next to those squares, we have a square with a side length of 5, thus with an area of 25.
5 x 5 = 25
Once we entered the Middle Ages, the concept of infinity was used on many different applications, specifically, the division of fractions. In the sixteenth century, Rafael Bombelli utilized the concept of infinity while also addressing the problem with imaginary numbers. He presented this in the square root of thirteen, by continually adding divided fractions
\sqrt{13}\approx3.60555127546by using the form
\sqrt{a^{2}+b}=a+\frac{b}{2a+{\frac{b}{2a+{\frac{b}{2a+{\frac{b}{2a+}....}}}}}} \sqrt{3^{2}+4}=3+\frac{4}{2(3)+{\frac{4}{2(3)+{\frac{4}{2(3)+{\frac{4}{2(3)+}....}}}}}}\approx3.60555By the 17th century, Galileo Galilei was using finite reasoning on infinite things. Galileo stated, “This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.” Galileo then presented his paradox, which shows that the list of perfect squares is as infinite as a list of numbers. As a result, for the list of perfect squares, we have 1, which is 1 squared, 4, which is 2 squared, 9 which is 3 squared, 16 which is 4 squared, 25, 36, 49, 56, 81, and on and on and on.
Galileo’s Paradox states that even though there are some counting (aka natural) numbers that when squared or cubed, are more numerous than other numbers, each of these numbers has a one-to-one correspondence with the squared or cubed value. As a result, there can’t be more numbers in those one-to-one sets because all the numbers in one set match up with the numbers in the other set.
The paradox is that in the second set, there are fewer natural numbers. In other words, there is a gap between 1 and 4, where 2 and 3 are missing. There is a gap between 4 and 9, where 5, 6, 7, and 8 are missing. However, the sizes of the sets are the same.
This process also works with number lines. For example, in the following image, we have a number line that starts with 0 and ends with 2. Below that is a number line that starts with 0 and ends with 4. We can show that there is a one-to-one correspondence between the smaller number line and the larger number line. The rule applied in the image below is that if we take every number in the upper number line and multiply it by 2, we get its corresponding number. In reverse, if we take every number in the lower line and divide it by 2, we get the corresponding value in the upper line.
What Galileo showed is that these sets of numbers can both equal infinity. As a result, Galileo showed that we cannot apply the concepts of equal, greater, and less than to the infinite. Those concepts can only be applied to finite quantities. And, this was the challenge. He had shown that in infinite quantities, an infinite list of squared numbers is larger than an infinite list of non-squared numbers. And so, he grappled with how to show that some infinite sets are larger than other infinite sets. He couldn’t do it. This task actually wasn’t accomplished until about 250 years later through the genius of Georg Cantor.
Georg Cantor was born in 1845. He was an exceptional violinist and a brilliant mathematician. Though he was born in Russia, he was a prominent German mathematician. Many of his peers and contemporaries thought he was crazy because he came up with this concept of set theory. It’s now a very valuable fundamental tool used in mathematics. But for Cantor, who first proposed it, he was ridiculed and insulted. Cantor introduced cardinal and ordinal numbers and proposed that there are different types of infinity. In his theorem, he implied the existence of an infinity of infinities. This concept led to his idea that there are transfinite numbers. Transfinite numbers are numbers that are infinite in such a way that they are larger than finite numbers; however, they are not absolutely infinite.
Using set theory, Cantor proved that there different sizes for infinite sets. For example, in one of his first papers, he proved that an infinite set of real numbers has more numbers than a set of natural (counting) numbers. Additionally, Cantor showed that the set of all real numbers is uncountable using a method that is now called the Cantor diagonalization process.
This process validated that finite sets are countable. In other words, a set with three numbers, 1, 2, and 3, is countable: {1,2,3}. And it has a cardinality of 3 since there are three finite numbers in the set.
|{1,2,3}|=3
It also validated that a countable set shares a one-to-one correspondence with the set of positive integers. In this case, the set of all positive integers, represented as
\Z^{+}has a cardinality of Aleph null, which is represented with
\aleph_{0}As a result
\Z^{+}=\aleph_{0}But Cantor also showed that there are infinite sets that do not have a one-to-one correspondence with an infinite set of natural numbers. As a result, an infinite set is not countable. What could go into this infinite set? The set of all real numbers. So, for example, between 0 and 1, there is an infinite amount of real numbers! You simply cannot list all the real numbers in a number line between 0 and 1.
In the Diagonalization process, Cantor pointed out that we cannot list all the real numbers by creating a set of decimals. Then he created a new number by taking the numbers listed in the diagonal. Then he created a new number from the number he grabbed in the diagonal by making one small change. For this example, the new rule will be to add 1 to each of those digits. As a result, he created an entirely new number that is not on the list. By doing this, he proved that all real numbers are an infinite set and are uncountable. This was a different type of infinity. And through this, Cantor showed that there is an infinite list of various sized sets of infinity. This was the foundation of set theory.
Cantor was a genius with original ideas, much like Einstein. Sadly, he was rebuked by his contemporaries. Henri Poincare, Leopold Kronecker, and others described him as a “scientific charlatan.” Kronecker went so far as to verbalize that he was uncomfortable working with Cantor, and even intentionally delayed the release of Cantor’s first publication. Every time Cantor would apply to work at the University of Berlin, the university would reject his application. Many believed that Kronecker had a say in these rejections. In other words, Kronecker was being a real jerk. Thus, Cantor remained at the University of Halle, where he worked during his entire career and made a modest income. Cantor’s contemporaries even accused him of corrupting the youth with his crazy ideas. Ludwig Wittgenstein even went so far as to say that his theories were laughable and wrong and complete utter nonsense.
Cantor began to struggle with ongoing bouts of depression. In 1884, he entered a sanatorium. It was believed that he might have had bipolar disorder. Thus, his depression worsened as his peers continually mocked him.
On December 16, 1899, while Cantor was giving a lecture on his views on Baconian Theory and William Shakespeare, Cantor’s 12-year-old son, Rudolph, died unexpectedly. This was a devastating turn of events for Cantor. He lost his passion for his life and his work. In 1903, he entered the sanatorium again for depression.
Even after this, his contemporaries continued to mock him. However, he also received many honors for his work. In 1904, the Royal Society awarded Cantor with the Sylvester Medal, which is the highest honor for work in mathematics. As Cantor’s critics continued to vilify his work, David Hilbert came to Cantor’s defense, stating, “From his paradise that Cantor with us unfolded, we hold our breath in awe; knowing, we shall not be expelled.”
However, that same year, in 1904, Julius Koenig presented a paper at the Third International Congress of mathematicians where he tried to prove that transfinite set theory was utterly false. Koenig read the paper in front of Cantor’s daughters and colleagues, which devastated Cantor. Even though Koenig’s proof of Cantor’s work failed, Cantor was still distressed and continued to struggle emotionally. By 1913, he was poor and malnourished. By 1917, he entered the sanatorium again. He had continually asked his wife if he could go home. However, she wouldn’t allow him to return. Then, on January 6, 1918, he had a heart attack and died.
It is a horribly sad ending for such a brilliant man who created fortifying foundations to set theory. The concept of infinity can be applied to so many areas today. One of the best explanations I’ve heard for infinity in physics comes from Hakeem Oluseyi in his Ted Talk.
In his talk, he describes an electron traveling through space. That electron can spontaneously become two photons, which then can spontaneously become a matter and an antimatter particle, which then can spontaneously become an ongoing, infinite sum of subatomic particles. This is infinity at its most tangible.
To quote Einstein again, he once said that imagination is more important than knowledge. Because of imagination, in the last 100 years, we have made technological leaps that we could never have accomplished in the nineteenth century. So, if we were to add imagination to infinity, consider all the infinite possibilities we could obtain. Imagine all the infinite possibilities for our mathematical advancements. Imagine all the infinite possibilities for our scientific advancements. Imagine all the possibilities in advancements for humanitarianism. I would like to hope that if we can imagine it, we can achieve it. But, like Zeno’s Paradox, it may take forever to get from here to there.