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Tales on Infinities

Author: Jean Claude Ted Isidor

Editors: Yanxi Chen and Ivan Feng

Artist: Serena Zhou

At first, the idea of infinity seems subtle and relatively meaningless. However, some might say, “infinity just means forever, like my math homework!” Perhaps others with a more philosophical approach to these questions might reply, “Infinities is a concept from mathematics at its finest, such a concept is too vast for our primitive brains to perceive,” which is a run-on sentence on the 'mystical' nature of infinities and filled with jargon. Mathematicians, on the contrary, view infinities with awe.

Some mathematicians, like the infamous David Hilbert, came up with fascinating stories utilizing infinities at its core. Hilbert’s story was based around a hotel with “infinitely many rooms,” which sounded quite like the Marriott hotels. This hotel, with an infinitely tired custodian, had reached its maximum capacity, holding an infinite number of guests all heading to Disney world. Another tourist booked a room, hoping to get some rest before hearing the joyful screams of a child as mickey mouse came to greet them. At first, the receptionist seemed quite anxious on what to do, as in the Hilbert hotel, no guest was turned away, but the sly looking manager in the back room simply grabbed the announcement speaker and said “all guests move one room to the right.” Suddenly a stampede of people in flower shirts and neon colored shorts were seen grabbing their luggage and hustling quickly to the adjacent room. Following this commotion the manager said “congratulations, room 1 is now vacant and has your name written on it,” and went back to slouching on his chairs.

It was confusing as to how a hotel with infinite rooms and holding an infinite amount of guests is able to just make room for one more, defying what would be considered its capacity. The humor in the answer was that in the Hilbert hotel scenario, we shifted all guests to the right, turned the amount of rooms from infinity to infinity plus one, keeping the capacity infinite. David Hilbert created a simple yet perplexing idea, in which the concept of infinity is no different from such a thought.

As with many other mathematical concepts, the idea of infinity can be traced back to the Greeks. However, they weren’t the only unorthodox mathematicians in the early ages of history. According to a paper titled "A Finite History of Infinity" by Amy Whinston, one of the Greeks original ideas was about cutting an object in half infinitely, for example, taking an orange and cutting it into two slices, and those two slices into halves of halves and so on. The ancient Greeks thought that if one had the energy to do such a task, they could continue to divide the orange indefinitely, reaching infinity. This idea was heavily debated amongst their greatest minds. Euclid, when proving the notion that there exists infinite primes, said that he did not prove the existence of primes being infinite but rather that the number of prime numbers cannot be measured. Much thought on infinities at the time were done by philosophers, who applied the notion of something being neverending to religious deities and the size of the universe. One such philosopher considered the effects of throwing a ball at the edge of the universe, where the ball can only be stopped if something outside of the universe interferes with it. The philosopher then went on to reason based on the idea that the universe is infinite and a considerable amount of people to this day believe so as well.

Galileo Galilei, commonly considered the first modern scientist, was the greatest contributor to mathematical advancement of the concept of infinities. The astronomer was one of the first to begin classifying infinities. In his work on infinities, Galileo discussed 2 of the numerous types of infinities: potential infinities and actual infinities. The list of natural numbers—integers starting from one—demonstrates infinity as striving to reach an impossible destination. A better explanation of potential infinities comes from the essay “Potential versus Completed Infinity'' by Eric Schecter, in which Schecter says that potential infinities are like infinitely long lines, with the end being infinity itself, using an “it seems to be going this way” as the indicator. Actual infinities use the notation of placing a set in rounded brackets, or rather simply a collection of things. {2,4,6,8,...} is considered as an actual infinity because of what it contains. By having the curly brackets, we say that the rest of the set, which is assumed to be {10,12,14,...}, is already inside of the set! Therefore, the set actually reaches infinity as it contains infinite numbers while potential infinites only suggests it. Continuing after, Galileo discovered calculus—which relied heavily on the notion of infinity. It’s creators, Newton and Leibniz, based the two important principles of derivation and integration on the notion of dividing a function into infinitely smaller pieces.

Infinity through calculus was now able to have an effect on applied mathematics instead of just philosophy and theoretical mathematics. The use of calculus in space exploration, manufacturing products, and creating video games have made infinity a concept we all indirectly encounter. Most people are also familiar with the infinity symbol, but not its origin. According to Whinston, the designer of the symbol for infinity, John Wallis, wanted a symbol that can be traced an infinite amount of time. As you traverse the horizontal eight again and again, take time to think about the wonders of infinity.

The use of infinities is still being expanded today. In the past year, two mathematicians, Maryan the Malliaris and Saharon Shelah, constructed the proof that two infinities p and t (with p being smaller than t) are actually equal. In set theory, two sets being similar in magnitude means that they have the same cardinality. The variables can then be substituted with numbers. The set of integers is {...-3,-2,-1,0,1,....} while the set of real numbers consists of {....-3.00021345,-3.000034,...}. If plotted on a number line, the set of real numbers would be every point on that line. Mathematicians prove the cardinality of sets through the one-to-one correspondence, in which elements from each set are paired with each other. Sets that have one to one correspondence like {1,2,3,4,5,..} and {5,10,15,20,…} (pairs being 1 and 5, 2 and 10, and so on) are called countable lists.

On the other hand, there exist sets that cannot be paired such as the set of integers and the set of real numbers mentioned earlier. If an attempt is made to map every integer to a real number, one would soon realize that no matter how cleverly it is mapped, there will always be additional real numbers, and therefore, the values of the infinities differ. This led Cantor, the creator of set theory, to speculate whether there might be another type of infinity, one that stands between countable and uncountable ones, though he deemed that it did not exist. This is important as the question of whether two infinities p and t are equal directly involves looking into this hypothesis. Through further study, many researchers concluded that this equality issue between different infinite sets could not be solved through set theory alone. Malliaris and Shelah had to dive into a newer section of mathematics called Kleir’s order, in which he “classified theories based on complexity.” Using this blueprint, the two mathematicians studied the complexity of the individual sets; the rest was history.

When discussing the concept of infinities or any concept in mathematics, it is incredible to see how far it has come and how it continues to grow. Mathematics is one of the best ways to see and experience collaboration, with people from different corners of the globe working together simply to find something, whatever it may be.

“Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.”

-David Hilbert

Citations:

“60+ Brilliant Math Quotes Every Teacher Needs to Read.” Prodigy Education, Game, Prodigy,

2019, https://www.prodigygame.com/main-en/blog/math-quotes/.

“Infinity: World of Mathematics.” Mathigon, https://mathigon.rog/world.Infinity.

“Schechter, Eric.” Potential vs Completed Infinity, Vanderbilt University, 5 Dec. 2009,

https://math.vanderbilt.edu/schectez/courses/thereals/potential.html.

“A Finite History of Infinity- Portland State University” A Finite History of Infinity, Whinston,

Amy, Portland State University, 2009,

https://web.pdx.edu/~caughman/AmyDraft501.pdf.