The hypothesis that the cardinality of the set of real numbers is equal to the cardinality of the set of integers.

The hypothesis that the cardinality of the set of real numbers is greater than the cardinality of the set of integers.

The hypothesis that the cardinality of the set of real numbers is less than the cardinality of the set of integers.

The hypothesis that the cardinality of the set of real numbers is equal to the cardinality of the set of rational numbers.

$\aleph_0$

$\aleph_1$

$\aleph_2$

$\aleph_3$

$\aleph_0$

$\aleph_1$

$\aleph_2$

$\aleph_3$

True

False

Undecidable

The hypothesis that the cardinality of the set of real numbers is equal to the cardinality of the set of integers.

The hypothesis that the cardinality of the set of real numbers is greater than the cardinality of the set of integers.

The hypothesis that the cardinality of the set of real numbers is less than the cardinality of the set of integers.

The hypothesis that the cardinality of the set of real numbers is equal to the cardinality of the set of rational numbers.

The Generalized Continuum Hypothesis implies the Continuum Hypothesis.

The Continuum Hypothesis implies the Generalized Continuum Hypothesis.

The two hypotheses are independent of each other.

None of the above.

Georg Cantor

David Hilbert

Kurt Gödel

Paul Cohen

Georg Cantor

David Hilbert

Kurt Gödel

Paul Cohen

It showed that the Continuum Hypothesis is true.

It showed that the Continuum Hypothesis is false.

It showed that the Continuum Hypothesis is independent of the axioms of Zermelo-Fraenkel set theory.

None of the above.

The Continuum Hypothesis is true.

The Continuum Hypothesis is false.

The Continuum Hypothesis is independent of the axioms of Zermelo-Fraenkel set theory.

The axioms of Zermelo-Fraenkel set theory are inconsistent.

It is true.

It is false.

It is independent of the axioms of Zermelo-Fraenkel set theory.

None of the above.

Is the Continuum Hypothesis true or false?

Can the Continuum Hypothesis be proven or disproven using a different set of axioms?

Are there other mathematical theories in which the Continuum Hypothesis can be proven or disproven?

All of the above.

It has implications for the foundations of mathematics.

It has applications in other areas of mathematics.

It is a challenging problem that has attracted the attention of many mathematicians.

All of the above.

In computer science, the Continuum Hypothesis could be used to study the complexity of algorithms.

In physics, the Continuum Hypothesis could be used to study the structure of space-time.

In economics, the Continuum Hypothesis could be used to study the behavior of markets.

All of the above.

Mathematicians will continue to search for a proof or disproof of the Continuum Hypothesis.

Mathematicians will explore new set theories in which the Continuum Hypothesis can be proven or disproven.

Mathematicians will investigate the applications of the Continuum Hypothesis in other areas of mathematics and science.

All of the above.