Final answer:
The Continuum Hypothesis is a mathematical hypothesis that addresses the size of infinity. It states that there is no set with a cardinality strictly between that of the set of natural numbers and the set of real numbers. It is an independent hypothesis that cannot be proven or disproven using the standard axioms of set theory.
Step-by-step explanation:
The Continuum Hypothesis is a mathematical hypothesis that addresses the size of infinity. It was first proposed by Georg Cantor in the late 19th century and is related to the concept of sets and their cardinality. The hypothesis states that there is no set with a cardinality strictly between that of the set of natural numbers (countable infinity) and the set of real numbers (uncountable infinity).
This hypothesis has been extensively studied in the field of set theory. In 1963, Paul Cohen showed that the Continuum Hypothesis cannot be proved or disproved within the widely accepted framework of set theory known as Zermelo-Fraenkel set theory. This result implies that the Continuum Hypothesis is independent of the standard axioms of set theory, meaning that it cannot be either proven or disproven using those axioms.
The Continuum Hypothesis is a deep and challenging problem in set theory, and mathematicians continue to explore its implications and connections to other areas of mathematics.