Say a wonderful thing! The continuum hypothesis can never be proved!

For the finite set A of elements, we can count the elements one by one, which is the size of set A, and it is denoted as |A|. But what about infinite sets?

Cantor thought of a way: if the elements of two infinite sets correspond to each other, then they are the same size. Ω is the set of all natural numbers (including 0). Choose it as the scale of comparison, so that и0 = | Ω |, and those greater than и 0 are set as и 1, и 2, ...

Cantor successively proved the integer set | z ||||||| ω| и 0 and the rational number set | q|||| ω| и 0, but found the real number set | r ||||||||||| ω|| и 0, that is | r | и 0. Then the question comes: which of |R| should be equal to и 1, 10802, ... Cantor guessed: | r | и 1.

Because the real number covers the whole line and is continuous, it is called continuous system, which is called continuum for short, and then this assumption is called continuum hypothesis.

The continuum hypothesis cannot be proved to be correct under ZFC (or its equivalent) axiomatic system, but neither can it be proved to be wrong.

Compatibility, completeness and computability of mathematics

It was rejected by Godel and Turing.

Euler, a great mathematician, put forward a conjecture that the equation X 4+Y 4+Z 4 = W 4 has no positive integer solution For more than 200 years, people can neither prove Euler's conjecture nor find a counterexample.

But in 1988, a mathematician from Harvard University discovered four integers, which directly falsified Euler's conjecture. These four numbers are:

2682440, 15365639, 18796760,206 15673。

(The sum of the fourth powers of the first three numbers is equal to the fourth power)