What Gödel Discovered?

What Gödel Discovered | Article | Abakcus
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In 1931, a 25-year-old Kurt Gödel wrote a proof that turned mathematics upside down. The implication was so astounding, and his proof so elegant that it was…kind of funny. I wanted to share his discovery with you. Fair warning though, I’m not a mathematician; I’m a programmer. This means my understanding is intuitive and not exact. Hopefully, that will come to our advantage since I have no choice but to avoid formality ?. Let’s get to it.

Unification

For the last 300 years, mathematicians and scientists alike made startling discoveries, which led to one great pattern. The pattern was unification: ideas that were previously thought to be disparate and different consistently turned out to be one and the same!

Newton kicked this off for physicists when he discovered that what kept us rooted on the Earth was also what kept the Earth dancing around the sun. People thought that heat was a special type of energy, but it turned out that it could be explained with mechanics. People thought that electricity, magnetism, and light were different, but Maxwell discovered they could be explained by an electromagnetic field.

Darwin did the same for biologists. It turned out that our chins, the beautiful feathers of birds, deer antlers, different flowers, male and female sexes, the reason you like sugar so much, the reason whales swim differently…could all be explained by natural selection.

Mathematicians waged a similar battle for unification. They wanted to find the “core” principles of mathematics, from which they could derive all true theories. This would unite logic, arithmetic, and so on, all under one simple umbrella. To get a sense of what this is about, consider this question: How do we know that 3 is smaller than 5? Or that 1 comes before 2? Is this a “core” principle that we take on faith (the formal name for this is called an “axiom”) or can this be derived from some even more core principle? Are numbers fundamental concepts, or can they be derived from something even more fundamental?

Crisis

Mathematicians made great progress in this battle for core principles. For example, a gentleman called Frege discovered that he could craft a theory of sets, which could represent just about everything. For numbers, for example, he could do something like this:

What Gödel Discovered

Here, he represents 0 the empty set. 1 as the set which contains the set for 0. 2 as the set that contains the set for 1 and 0. From this he could set a principle to get the “next” number: just wrap all previous numbers in a set. Pretty cool! Frege was able to take that and prove arithmetic rules like “1 + 1”, “numbers are infinite”, etc.

This looked formidable and cool, but Bertrand Russell came along and broke the theory in one fell swoop.

He used the rules that Frege laid out to make a valid but nonsensical statement. 

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