For many years, people believed that there are the same number of natural numbers and square numbers. This is because Galileo Galilei proved that if you lined up each natural number with its square, all the way up to infinity, you would end up with the same number of numbers. In 1875, Georg Cantor wondered if there were two infinite sets that didn’t match up. Would there then be different infinities?
To figure this out, he lined up real numbers and the natural numbers between 0 and 1. Because there is no smallest natural number between 0 and 1, Cantor just listed them at random, using a method to make sure that no two numbers are the same. Through this process, he discovered that there are more natural numbers between 0 and 1 than real numbers; infinity comes in different sizes. Shortly after, Cantor classified some infinities like square numbers and integers into countable infinities. Other infinities (like set of all real numbers) were classified into uncountable infinities. They can’t exactly be matched to the natural numbers. After Cantor published his discovery, many people were astonished. After all, how can something that goes on forever be bigger than something else that goes on forever?
Cantor then wanted to see if infinite sets could also be placed in a specific order. He used the Well Ordering Principle for this. The two conditions for a set to be well-ordered are that the set must have a clear starting point and that the subset (a set in another larger set) must also have a clear starting point. He then introduced his work to the public in 1883, but because he couldn’t mathematically prove his studies, the math community attacked Cantor.
In 1904, Julius König announced that he had proof that Cantor’s theory was wrong. In the audience was Cantor, his wife, and his children. There was someone else who was also in the audience, Ernst Zermelo. Zermelo figured out a pattern, in his words, “which Cantor uses unconsciously and instinctively everywhere, but formulates explicitly nowhere.” Because axioms are rules in math that we accept without proof, Cantor just needed to add something to make his assumption hold up in a system of proofs.
Zermelo came up with the Axiom of Choice to help support Georg Cantor’s idea. The Axiom of Choice created many paradoxes (a statement or situation that may seem contradictory), the most famous being the Banach-Tarski paradox. It states that a sphere can be decomposed into infinite pieces. From those pieces you can then form two identical copies of the original sphere.
In 1938, Kurt Gödel proved that the Axiom of Choice was true. In 1963, Paul Cohen proved that in the world the other axioms proved true, but the Axiom of Choice didn’t. After that, there wasn’t a lot of debate about the Axiom of Choice.
Related Stories:
https://jaydaigle.net/blog/what-is-the-axiom-of-choice/
https://en.wikipedia.org/wiki/Axiom_of_choice
https://plato.stanford.edu/entries/axiom-choice/
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