The Basics of Infinity
Infinity is often thought of as something that goes on forever. For example, the set of natural numbers (1, 2, 3, 4, ...) is infinite because you can always add one more to any number in the set. This type of infinity is known as "countable infinity" because you can, in theory, count the elements one by one, even if it takes forever. The notation for the size of this set is ℵ₀ (aleph-null).
Countable vs. Uncountable Infinity
Georg Cantor's groundbreaking insight was that not all infinities are the same size. He demonstrated this by comparing the set of natural numbers (countable infinity) with the set of real numbers (uncountable infinity). Countable infinity refers to sets like the natural numbers. You can list them out one by one, and even though the list never ends, each element can be matched with a unique natural number. In contrast, the set of real numbers between 0 and 1 is uncountably infinite. To understand why, Cantor used a method called the "diagonal argument."
Cantor's Diagonal Argument
Cantor's diagonal argument shows that there are more real numbers between 0 and 1 than there are natural numbers, meaning that the set of real numbers is uncountably infinite. Suppose we try to list all the real numbers between 0 and 1. Imagine writing them in an infinite list, where each number is represented by its decimal expansion. For example, consider a list like this:
0.123456...
0.987654...
0.543210...
0.461323...
Cantor showed that no matter how you list these numbers, you can always create a new number that is not on the list by changing the nth digit of the nth number in your list. For instance, if the first digit of the first number is 1, change it to 2; if the second digit of the second number is 9, change it to 8, and so on. This new number will differ from every number in the list in at least one digit, meaning it cannot be part of the original list.
This proves that the set of real numbers is uncountable because there is no way to list all of them without missing some.
Cantor's Legacy
Georg Cantor's work on different sizes of infinity was revolutionary. He developed the concept of "cardinality" to compare the sizes of infinite sets and introduced the idea that some infinities are larger than others. His work laid the foundation for much of modern set theory and had a profound impact on mathematics.
Cantor faced significant opposition from his contemporaries, many of whom found his ideas too radical. Despite the criticism, Cantor remained committed to his theories and continued to develop his ideas, which are now widely accepted and celebrated in the mathematical community.
Conclusion
The concept of different-sized infinities is a fascinating and counterintuitive aspect of mathematics. Thanks to Georg Cantor, we now understand that infinity is not a one-size-fits-all concept but rather a rich and complex idea with many layers. Next time you ponder the infinite, remember that there are infinities larger than others and that our journey into the world of mathematics is as endless as the numbers themselves.
Work Cited
Matson, John. “Strange but True: Infinity Comes in Different Sizes.” Scientific American, https://www.scientificamerican.com/article/strange-but-true-infinity-comes-in-different-sizes/. Accessed 6 June 2024.
Georg Cantor | Biography, Contributions, Books, & Facts | Britannica. 3 June 2024, https://www.britannica.com/biography/Georg-Ferdinand-Ludwig-Philipp-Cantor. Accessed 6 June 2024.
Cantor’s Theorem | Set Theory, Cardinality, Countability | Britannica. https://www.britannica.com/science/Cantors-theorem. Accessed 6 June 2024.
