Paul Cohen facts for kids

Paul Joseph Cohen (born April 2, 1934 – died March 23, 2007) was a brilliant American mathematician. He is most famous for solving a very difficult problem in mathematics called the continuum hypothesis. He also showed that another idea, the axiom of choice, could not be proven using the standard rules of set theory. For this amazing work, he received the Fields Medal, which is like the Nobel Prize for mathematics.

Early Life and Education

Paul Cohen was born in Long Branch, New Jersey, USA. His family had moved to the United States from what is now Poland. He grew up in Brooklyn, a part of New York City. He was a very smart student and graduated from Stuyvesant High School in New York City in 1950, when he was only 16 years old.

After high school, Paul went to Brooklyn College from 1950 to 1953. He was so good at math that he was able to start his graduate studies at the University of Chicago after only two years of college. At Chicago, he earned his master's degree in mathematics in 1954. He then completed his PhD in 1958. His main teacher there was Antoni Zygmund.

In 1957, Paul Cohen started working as a math instructor at the University of Rochester. He then spent time at the Massachusetts Institute of Technology and the Institute for Advanced Study at Princeton. During these years, he made important discoveries in math. For example, he solved a problem about how certain math functions can be broken down into simpler parts. He also made a big step forward in solving the Littlewood conjecture, another tough math problem.

Paul Cohen became a member of several important groups, like the American Academy of Arts and Sciences and the United States National Academy of Sciences. In 1995, he received an honorary doctorate degree from Uppsala University in Sweden.

Amazing Work in Mathematics

Paul Cohen is well-known for creating a new math method called forcing. He used this method to show that the continuum hypothesis (CH) and the axiom of choice cannot be proven from the usual rules of set theory. Set theory is a branch of math that studies collections of objects.

This meant that these ideas are "independent" of the standard rules. You can't prove them true or false using those rules alone. The continuum hypothesis is one of the most famous examples of a math problem that cannot be decided using standard axioms.

For his breakthrough on the continuum hypothesis, Cohen was awarded the Fields Medal in 1966. He also received the National Medal of Science in 1967. The Fields Medal he won is still the only one ever given for work in mathematical logic.

Besides his work in set theory, Cohen also made many valuable contributions to a field called analysis. He won the Bôcher Memorial Prize in 1964 for his paper on a problem by John Edensor Littlewood. His name is also part of the Cohen–Hewitt factorization theorem.

Paul Cohen was a full professor of mathematics at Stanford University. He was also invited to speak at major international math conferences in 1962 and 1966.

A famous mathematician named Angus MacIntyre once said that Cohen was "dauntingly clever." He compared Cohen's work to that of Kurt Gödel, another very famous mathematician, saying, "Nothing more dramatic than their work has happened in the history of the subject." Gödel himself wrote a letter to Cohen in 1963, praising his proof. He said it was "a delight to read" and "the best possible proof."

Understanding the Continuum Hypothesis

The continuum hypothesis was a very difficult problem that many mathematicians thought was impossible to solve. Paul Cohen once said in 1985 that people thought the problem was "hopeless."

The problem is about comparing the "size" of different infinite sets. Imagine counting numbers: 1, 2, 3... This is an infinite set. But what about all the points on a line? That's also an infinite set, but it seems "bigger" or "denser." The continuum hypothesis asks if there's any infinite size between the size of counting numbers and the size of all points on a line.

Cohen believed that the set of all points on a line is "incredibly rich." He thought it was much larger than other infinite sets that are built up step-by-step. He felt that later generations would understand this idea even better.

Cohen's method of "forcing" became a very important tool. It helps mathematicians build special mathematical models. These models can then be used to test if a certain idea or hypothesis is true or false. This method is still used by many mathematicians today.

Just before he passed away, Cohen gave a lecture in 2006. He talked about how he solved the continuum hypothesis problem. This was at a conference celebrating 100 years since Kurt Gödel's birth.

Death

Paul Cohen and his wife, Christina, had three sons. Paul Cohen passed away on March 23, 2007, in Stanford, California. He had been suffering from a lung disease.

See also

In Spanish: Paul Cohen para niños

• Cohen algebra