Continuum Hypothesis
Hypothesis
- Continuum Hypothesis: There does not exist a cardinal number $x$ that satisfies $\aleph_{0} < x < 2^{\aleph_{0}}$ for given $\aleph_{0} = |\mathbb{N}|$.
- Generalized Continuum Hypothesis: For any infinite cardinal number $a = |A|$, there does not exist a cardinal number $x$ that satisfies $a < x < 2^{a}$.
Explanation
Cantor demonstrated that not all infinities are equal using methods similar to the diagonal argument. He showed that even though infinite sets may exist, their cardinal numbers can be compared, and that the sets of natural numbers $\mathbb{N}$, integers $\mathbb{Z}$, and rational numbers $\mathbb{Q}$ are equivalent to each other, but not equivalent to the set of real numbers $\mathbb{R}$.
However, our familiar numerical system, that is $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$, might just be arranged in the order discovered by Westerners. No mathematics book states that $\mathbb{Q}$ is followed by $\mathbb{R}$. Regardless of the significance to humanity, there always exists some set that is slightly larger than $\mathbb{Q}$ and slightly smaller than $\mathbb{R}$. For example, adding only transcendental numbers to $\mathbb{Q}$ would create such a set. The question then becomes ‘what would be its cardinal number?’ This is precisely the Continuum Problem.
This problem was so challenging that it was listed among Hilbert’s 23 problems. The hypothesis that $x$ does not exist is known as the Continuum Hypothesis, and the generalization of this hypothesis to not just the set of natural numbers but to general sets is known as the Generalized Continuum Hypothesis.
The continuum problem was later resolved by showing that ‘whether true or false, no contradiction arises’, meaning ‘whether true or false, proving it is meaningless’. To emphasize once more, it was proved that it is okay whether $x$ exists or does not, rather than proving the existence or non-existence of $x$.