Continuum Hypothesis
Overview
The Continuum Hypothesis (CH) is a statement in the field of set theory, a branch of mathematical logic, that concerns the possible sizes of infinite sets. Proposed by German mathematician Georg Cantor in 1878, the hypothesis states that there is no set of cardinality strictly between that of the integers and the real numbers.
Historical Background
The Continuum Hypothesis was first proposed by Cantor in the late 19th century. Cantor's work in set theory and his development of the concept of cardinality - the measure of "number of elements in a set" - led him to the question of the size of the set of all real numbers. He hypothesized that the cardinality of the real numbers (the "continuum") is the next size up from the cardinality of the integers, hence the name "Continuum Hypothesis".
Statement of the Hypothesis
In terms of cardinal numbers, the Continuum Hypothesis can be stated as follows: If we denote the cardinality of the integers as ℵ0 (aleph-null) and the cardinality of the real numbers as c, then the Continuum Hypothesis states that there is no cardinal number which is between ℵ0 and c. In other words, the hypothesis asserts that ℵ1 = c.
Independence from Zermelo-Fraenkel Set Theory
In the mid-20th century, mathematicians Kurt Gödel and Paul Cohen showed that the Continuum Hypothesis cannot be proven or disproven from the axioms of Zermelo-Fraenkel set theory (ZFC), the most widely accepted foundational system for mathematics. Gödel showed in 1940 that the negation of the Continuum Hypothesis cannot be proven from the ZFC axioms, and Cohen showed in 1963 that the Continuum Hypothesis itself cannot be proven from the ZFC axioms. This means that the Continuum Hypothesis is independent of ZFC set theory.
Implications and Consequences
The independence of the Continuum Hypothesis from ZFC set theory has profound implications for the philosophy of mathematics and our understanding of the infinite. It shows that certain basic questions about the infinite cannot be answered on the basis of the commonly accepted axioms for set theory.