Let us play a game. "Largest Number Wins". Whoever can name the larger number than the other player wins that round and this game goes on for as many rounds as needed till someone loses a round. In short, higher number means winner. In this thought experiment, now consider who would win and when the game would end. You could keep naming numbers like "ten, million, trillion, googol, googolplexian, and so on". The game could never end... unless one "smart" participant proudly claims "INFINITY" and wins the game. But have they actually won?
Infinity is a concept that has puzzled mathematicians and thinkers for thousands of years. It is an abstract and fascinating concept that has played a crucial role in the development of many mathematical fields, from calculus to quantum physics.
To begin, let us define what infinity means. Infinity is a concept. Emphasis on concept. Essentially, it refers to a quantity that is endless or without bounds. It is not a number, but rather an idea of a limitless quantity that goes beyond what can be counted or measured. The concept of infinity has been used by mathematicians since ancient times to describe various aspects of the world, from the stars in the sky to the smallest particles that make up matter. So, no, that friend did not win the game.
History of Infinity
The concept of infinity has a long and rich history, dating back to ancient Greece. The first proposition of infinity is attributed to Anaximander, who believed that the universe was infinite and eternal. The concept of infinity was further developed by Aristotle, Zeno, and many Greek mathematicians. Its evolution over centuries through various mathematicians around the world culminated in the work of Georg Cantor. It can be considered that he was one of the first people to make infinity "formal" and truly define its properties. While introducing the concept of different sizes of infinity, he even proposed the sizes of infinity were infinite.
This claim seems straightforward. But give it a thought. What does it mean by the size of infinity being infinite? Somehow, Georg was suggesting we could compare infinities... Like one apple is more than two apples, he proposed one infinity was "more" than another. Odd, right? Wasn't the entire point of infinity that it could not be surpassed in value?
Hence, he was met with skepticism from both the communities of mathematics and philosophy. Most believing it to be too abstract and lacking in real-world applications made the sizes of infinity into a religious debate where theologists claimed the idea of multiple infinities overruled God’s “supreme control of infinity”. The situation only calmed over time as the world globalised and mathematicians moved from conservative mindsets to collaboration. The importance of infinity in mathematics, physics, and computing became increasingly apparent, and the concept of infinity became widely accepted and embraced by the mathematical community.
Types and Sizes of Infinity
There are two main types of infinity: countable and uncountable.
As the name suggests, countable infinity refers to the infinity that can still be "counted". Maybe not in actuality because sitting down and counting 1, 2, 3, 4.... to infinity is impossible. But, mathematicians formalised the definition as "a set of objects that can be put into one-to-one correspondence with the set of natural numbers". Let us break this definition down. We use the natural numbers to count (0, 1, 2, 3, 4…), hence any set where each of its objects can be assigned a counting number (0, 1, 2, 3, 4…) is countable. For instance, a set of 12 books is countable because each book can be assigned a number (1, 2, …. 12). Similarly, we expand this idea for the sets of numbers. For example, the set of even numbers is countably infinite because we can pair each even number with a natural number:
In contrast, the set of real numbers is uncountably infinite because it is impossible to put them into one-to-one correspondence with the natural numbers. (View Suggested Reading for why?). Take a second to understand what this means. Even numbers were infinite, yes, but countably infinite. Prime numbers are also infinite yes, but they are also countably infinite. Before, we draw amusing conclusions from this, let us define just one more term : cardinality.
The cardinality of a set refers to the size or magnitude of it. The cardinality of the set of factors of 6 {1, 2, 3, 6} is 4 because it has four terms in it. This concept of cardinality applies to infinity as well. There are different levels of infinity, each one larger than the previous one. The smallest level of infinity is Aleph-null (ℵ0), which refers to the cardinality of the set of natural numbers. So, in some sense, "there are ℵ0 natural numbers". Of course, take that statement with a grain of salt because you really cannot quantify the number of natural numbers. Then follows Aleph-one (ℵ1), two... These numbers represent "sets larger than the natural numbers". The concept of cardinality is important in set theory, which studies the properties of sets and their subsets. Now, we can make some fascinating statements:
The cardinality of natural numbers (0, 1, 2…) is the same as the cardinality of integers (…-2, -1, 0, 1, 2…)
The cardinality of natural numbers (0, 1, 2…) is the same as the cardinality of primes (2, 3, 5…)
The cardinality of real numbers between 0 and 1 (such as 0.78, 0.576, etc.) is the same as the cardinality of real numbers between 0 and 100 (such as 90.284, 46.392, etc.). Take a moment to understand what this statement is trying to say. We are saying that there as many numbers between 0 and 1 as there are between 0 and 100 or 0 and a million
Uses of Infinity
Infinity’s usage is expansive from the obvious applications in math and logic to fields like physics and computing.
Mathematics
One of the most important applications of infinity is in calculus, which is the study of rates of change. Limits, derivatives, and integrals are all concepts that involve infinity. Limits are used to describe the behavior of a function as the input approaches a certain value, which can be infinity. Let us take one basic example of the same:
Derivatives and integrals are used to find the slope and area of curves, respectively, which can also involve infinity. Infinite series are also important in series and progressions, which involves adding up an infinite number of terms.
Set theory is another field of mathematics that heavily relies on infinity. It studies the properties of sets and their relationships to each other. The concept of infinity is essential in set theory, which studies infinite sets and their properties. The study of infinite sets has led to many interesting and important results, such as the discovery of different sizes of infinity and the paradoxes of set theory. As you may have noticed, the various discussion about the sizes and cardinalities of infinites were in the context of "sets" of infinite objects. This should demonstrate the importance of infinity in set theory and vice versa.
Physics
Infinity also has important applications in physics. In physics, infinity is often used to describe the properties of space, time, and matter. For example, black holes are regions of space where the gravitational pull is so strong that not even light can escape. Black holes have infinite density and are infinitely curved, which makes them a fascinating subject for physicists. The concept of infinity is also important in the study of space-time, which is the four-dimensional space in which events occur in the universe. The theory of relativity, which is a cornerstone of modern physics, involves the concept of infinity in its description of the behaviour of objects in space and time.
Quantum mechanics is another field of physics that heavily relies on the concept of infinity. Quantum mechanics describes the behavioir of particles on a subatomic level, where typical physics no longer explains much. Quantum mechanics involves the use of complex equations and situations, which can take on infinite values, and the concept of infinity is essential in understanding the properties of quantum particles.
Conclusion
Infinity is a fascinating and complex concept that has played a crucial role in the development of many mathematical fields. Its usage is at different levels of complexity from simple, everyday mentions of it to intensive, detailed study about its nature. The origins and discoveries of infinity reflect the history of human thought. Where society and community initially found it difficult to break norms to accept infinity, the figure eight symbol of infinity has today become a ubiquitous symbol in mathematics and pop culture. Hence, the concept of infinity has captivated mathematicians and thinkers for centuries and will continue to do so for generations to come with its intricacies and thought-provoking research.
Bibliography and Suggested Further Reading
DeMichele, Thomas. “There Are Different Types of Infinity - Fact or Myth?” Fact / Myth, 14 Jan. 2019, factmyth.com/factoids/there-are-different-types-of-infinity/.
Hartnett, Kevin. “Mathematicians Measure Infinities, Find They’re Equal.” Quanta Magazine, 12 Sept. 2017, www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/.
India Today Web Desk. “Know about John Wallis, the Designer of Infinity Symbol.” India Today, 23 Nov. 2018, www.indiatoday.in/education-today/gk-current-affairs/story/know-about-john-wallis-the-designer-of-infinity-symbol-1394707-2018-11-23.
University of Victoria, and Gary MacGillivray. Chapter 7 : Cardinality of Sets. University of Victoria, pp. 3–5, web.uvic.ca/~gmacgill/LFNotes/Cardinality.pdf. Heading 7.3 : Cardinality of Infinite Sets.
Comments