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In the late nineteenth century, a German mathematician named Georg Cantor blew everybody's minds via understanding that infinities come in several sizes, called cardinalities. He proved the foundational theorems about cardinality, which modern day math majors have a tendency to be told in their Discrete Math categories.

Cantor proved that the set of real numbers is larger than the set of natural numbers, which we write as |ℝ|>|ℕ|. It was easy to establish that the size of the natural numbers, |ℕ|, is the primary endless measurement; no infinite set is smaller than ℕ.

Now, the true numbers are larger, but are they the second endless measurement? This turned out to be a far more difficult question, known as The Continuum Hypothesis (CH).

If CH is correct, then |ℝ| is the second one countless size, and no countless sets are smaller than ℝ, but larger than ℕ. And if CH is fake, then there may be a minimum of one measurement in between.

So what's the solution? This is the place things take a turn.

CH has been proven unbiased, relative to the baseline axioms of math. It may also be true, and no logical contradictions practice, but it surely will also be false, and no logical contradictions will observe.

It's a peculiar scenario, however not totally unusual in modern math. You could have heard of the Axiom of Choice, some other unbiased statement. The evidence of this end result spanned a long time and, naturally, break up into two main portions: the evidence that CH is constant, and the proof that the negation of CH is constant.

The first half is due to Kurt Gödel, the mythical Austro-Hungarian logician. His 1938 mathematical construction, known as Gödel's Constructible Universe, proved CH compatible with the baseline axioms, and continues to be a cornerstone of Set Theory categories. The 2nd part was once pursued for 2 more a long time until Paul Cohen, a mathematician at Stanford, solved it through inventing a whole approach of evidence in Model Theory referred to as "forcing."

Gödel's and Cohen's halves of the proof each take a graduate level of Set Theory to means, so it's no wonder this distinctive story has been esoteric outdoor mathematical circles.