Continuum hypothesis

The continuum hypothesis is a mathematical theory that concerns the nature of infinity. The hypothesis was first developed in the 1870s by German mathematician Georg Cantor. In its most basic form, the hypothesis attempts to answer the question of how many points there can be on a line. While one might assume the answer would be an infinite number, Cantor’s work showed that there can be more than just one type of infinity. He proposed that the infinity of cardinal numbers—whole numbers used to count, such as 1, 2, 3, 4, 5, and so on—is smaller than the infinity of real numbers—fractional numbers that can contain decimals, 1.5, 66.777, or the value of pi (3.14159...). Cantor’s continuum hypothesis states that the difference between the infinity of whole numbers and the infinity of real numbers is not infinite. Some mathematicians consider the hypothesis impossible to prove, while others see it as impossible to disprove. In 1900, a prominent mathematician included the continuum hypothesis on a list of twenty-three supposedly “unsolvable” mathematical problems.

Background

Georg Cantor was born on March 3, 1845, in St. Petersburg, Russia. His father was a merchant who worked in the St. Petersburg financial exchange. Cantor’s maternal grandfather had been a musician and played in the Russian orchestra. When Cantor was eleven, his family moved to Germany. As a young man, Cantor showed great musical talent and played the violin. However, he also became interested in mathematics and studied at the University of Zürich before moving to the University of Berlin in 1863. His academic focus includes mathematics along with physics and philosophy. After completing his dissertation in 1867, he taught briefly at a Berlin girl’s school before gaining a position as a lecturer at the University of Halle. He was named an assistant professor at the university in 1872 and a full professor in 1879.

Overview

Cantor’s background in mathematics and philosophy triggered his deep interest in the theory of numbers. He wrote ten papers on the subject between 1869 and 1873. Prior to his work, mathematics was mostly confined to the study of finite numbers, with the concept of infinity relegated to the realm of philosophy.

In 1874, Cantor began developing his concept of set theory, an idea that views mathematical sets as single units made up of individual components. Cantor defined a set as a collection of distinguishable objects thought of as a whole. Two sets are considered equal only if their individual elements are the same. When Cantor applied his idea of set theory to infinite numbers, he encountered a paradox that seemed to indicate one set of infinite numbers could be larger than another.

For example, when comparing a series of all numbers (1, 2, 3, 4, etc.) with only even numbers (2, 4, 6, 8, etc.), it may seem that the series of even numbers would be smaller. However, both sets of numbers can be paired on a one-to-one basis, such as 1 with 2, 2 with 4, 3 with 6, and so on. Both sets of numbers are the same size and can continue on until infinity.

However, comparing the cardinal, or whole numbers, such as 1, 2, 3, with real numbers such as 1.00, 2.00, 3.00, and so on, does not yield such a one-to-one match. For instance, pairing 1 with 1.00 and 2 with 2.00 leaves out an infinite amount of real numbers between them, such as 1.01, 1.001, 1.0001, 1.00001, and so on. Cantor’s work showed that it was impossible to assign each whole number to a real number, in effect, proving that the infinity of real numbers is larger than the infinity of whole numbers.

In mathematics, cardinality refers to the amount of elements in a grouping. For example, in the group red, yellow, and blue, the cardinality would be three, representing the three colors. Cantor used the Hebrew letter aleph (ℵ) to represent the cardinality of numbers, with ℵ0 representing the infinity of whole numbers. For other kinds of infinity, Cantor used the designations ℵ1, ℵ2, and so on.

Cantor’s continuum hypothesis stated that the difference between the infinity of whole numbers and the infinity of real numbers was not itself infinite. In this view, the infinity of whole numbers (ℵ0) was the smallest possible infinity, while the infinity of real numbers (ℵ1) was the largest. However, if the hypothesis was false, that meant there was at least another set of real numbers larger than the set of whole numbers but smaller than the final set of real numbers. In this case, the cardinality of real numbers would be redesignated ℵ2, ℵ3, or higher.

Cantor endeavored to solve his continuum hypothesis during his lifetime but was never successful. In 1900, German mathematician David Hilbert compiled twenty-three supposedly unsolvable mathematical problems and put Cantor’s continuum hypothesis atop the list. Of the twenty-three problems, ten have been conclusively solved by 2023, with seven more considered partially solved or whose proposed solution has been disputed.

The continuum hypothesis is among those considered to have been partially solved, although to a very limited extent. In the 1930s, Austrian mathematician Kurt Gödel applied a recently proposed mathematical framework of set theory to the problem. Gödel’s work did not provide a contradiction to Cantor’s, meaning that he had shown the continuum hypothesis could not be proven false by conventional mathematics.

In the 1960s, American mathematician Paul Cohen developed a new mathematical framework in an attempt to solve the continuum hypothesis. Cohen’s work showed that the continuum hypothesis could not be proven true. Gödel and Cohen’s results seem to indicate that the continuum hypothesis can neither be proven true nor false by known mathematical models.

Bibliography

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Wolchover, Natalie. “How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.” Quanta Magazine, 15 July 2021, www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/. Accessed 3 Apr. 2023.