Gödel’s First Incompleteness Theorem
You probably think that math is a purely logical, 100% consistent, black and white subject. Everything in math is either right or wrong: 1 +1 has only one correct answer, the solutions of 5x² + 6x + 1 = 0 are either 1 and -2.2 or they are not, and the Goldbach Conjecture can be only proven either true or false. For almost two thousand years, since the Ancient Greeks, this is the way math was practiced, by noble scholars and the mass population alike. It was believed by all the great mathematicians, from Aristotle to Newton, that every true statement has a proof. Even David Hilbert himself, while publishing his famous 23 greatest unsolved problems, stated, “Every mathematical problem should have a solution.” The proof may be hard to find, such as how the proof of Fermat’s Last Theorem took 350 years, but regardless of how hard a problem is, a solution always exists in black and white logic. It was believed that anything that we don’t know in the realm of mathematics, we don’t know because of our human incapability to solve the problem.
Then came the continuum hypothesis, proposed by German mathematician Georg Cantor in 1878. Just like many math problems before it, the CH had no clear solution. Many mathematicians devoted hundreds of hours to this problem, yet no one was able to either prove or disprove the hypothesis. To show you how big of a deal this problem is, the continuum hypothesis was listed first on Hilbert’s greatest unsolved problems of the 19th century.
The continuum hypothesis states, “There is no set whose cardinality is strictly between that of the integers and the real numbers.” (Don’t worry, I’ll explain what this means later.)
In 1940, Austrian mathematician Kurt Gödel demonstrated that the continuum hypothesis cannot be proven false. Then, in the 1960s, U.S. mathematician Paul Cohen showed that the continuum hypothesis cannot be proven true. Taken together, this meant that the continuum hypothesis cannot be answered??¿¿