# Continuum hypothesis

### 1878

It states: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^{\aleph_0}=\aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23&nbsp;problems presented in 1900.

### 1900

It states: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^{\aleph_0}=\aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23&nbsp;problems presented in 1900.

It became the first on David Hilbert's list of important open questions that was presented at the International Congress of Mathematicians in the year 1900 in Paris.

### 1923

This viewpoint was advanced as early as 1923 by Skolem, even before Gödel's first incompleteness theorem.

### 1931

An immediate consequence of Gödel's incompleteness theorem, which was published in 1931, is that there is a formal statement (one for each appropriate Gödel numbering scheme) expressing the consistency of ZFC that is independent of ZFC, assuming that ZFC is consistent.

### 1940

This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term the continuum for the real numbers. ==History== Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it.

Axiomatic set theory was at that point not yet formulated. Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory.

### 1963

This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term the continuum for the real numbers. ==History== Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it.

The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set – was proved in 1963 by Paul Cohen. ==Cardinality of infinite sets== Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspondence) between them.

### 1966

Cohen was awarded the Fields Medal in 1966 for his proof. The independence proof just described shows that CH is independent of ZFC.

### 1976

American Mathematical Society, 1976, pp.&nbsp;81–92.

### 1986

In 1986, Chris Freiling presented an argument against CH by showing that the negation of CH is equivalent to Freiling's axiom of symmetry, a statement derived by arguing from particular intuitions about probabilities.

### 2000

Hugh Woodin has attracted considerable attention since the year 2000.

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