# What are the useful outcomes of denying the Continuum Hypothesis?

## Why is continuum hypothesis important?

The continuum hypotheses (CH) is one of the most central open problems in set theory, one that is important for both mathematical and philosophical reasons. The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory.

## Is the continuum hypothesis unsolvable?

The continuum hypothesis is a problem of a very different kind; we actually can prove that it is impossible to solve it using current methods, which is not a completely unknown phenomenon in mathematics.

## Why is the continuum hypothesis Undecidable?

Together, Gödel’s and Cohen’s results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the axiom of choice).

## Why is the continuum hypothesis independent?

The independence of the continuum hypothesis (CH) from Zermelo–Fraenkel set theory (ZF) follows from combined work of Kurt Gödel and Paul Cohen. Gödel showed that CH cannot be disproved from ZF, even if the axiom of choice (AC) is adopted (making ZFC).

## What if the continuum hypothesis is false?

If the continuum hypothesis is false, it means that there is a set of real numbers that is bigger than the set of natural numbers but smaller than the set of real numbers. In this case, the cardinality of the set of real numbers must be at least א‎2.

## What is the continuum theory?

Continuum Theory is the study of compact, connected, metric spaces. These spaces arise naturally in the study of topological groups, compact manifolds, and in particular the topology and dynamics of one-dimensional and planar systems, and the area sits at the crossroads of topology and geometry.

## Is the continuum hypothesis equivalent to the axiom of choice?

The General Continuum Hypothesis implies the Axiom of Choice. Assuming the General Continuum Hypothesis, we will derive the Axiom of Choice in its equivalent version that every infinite cardinal is an aleph. First we establish three lemmas.

## What is Cantors continuum hypothesis?

Georg Cantor’s continuum hypothesis states that there is no cardinal number between ℵ0 and 20. In 1940 Kurt Gödel had shown that, if one accepts the Zermelo-Fraenkel system of axioms for set theory, then the continuum hypothesis is not disprovable.