Paul Cohen

Definition

Paul Cohen was a prominent American mathematician known for his groundbreaking work in set theory and logic, particularly in demonstrating the independence of the Continuum Hypothesis from Zermelo-Fraenkel set theory with the Axiom of Choice. His innovative method of forcing transformed how mathematicians approached the foundations of set theory and significantly influenced subsequent research directions in the field.

5 Must Know Facts For Your Next Test

1. Cohen received the Fields Medal in 1966 for his work on the independence of the Continuum Hypothesis, marking a major milestone in mathematical logic.
2. His method of forcing revolutionized set theory by allowing for the creation of models where certain propositions could hold true, showcasing a new perspective on set-theoretic independence.
3. Cohen's work not only addressed the Continuum Hypothesis but also opened doors to exploring other propositions' independence from standard set theory axioms.
4. He contributed significantly to understanding large cardinals and their properties, further deepening the exploration of infinity in set theory.
5. Cohen's legacy continues to impact contemporary research directions in set theory, influencing how mathematicians investigate foundational questions and assumptions.

Review Questions

• How did Paul Cohen's method of forcing change the landscape of set theory?
  • Cohen's method of forcing introduced a powerful technique for constructing models of set theory where various propositions could be shown to hold or fail. This ability to manipulate and explore different models allowed mathematicians to demonstrate the independence of numerous hypotheses, including the Continuum Hypothesis. By doing so, he reshaped how foundational questions were approached and highlighted the complexity of infinite sets.
• In what ways did Cohen's work on the Continuum Hypothesis interact with Gödel's findings in set theory?
  • Cohen's work complemented Gödel's earlier results regarding the Continuum Hypothesis by showing that while it is consistent with Zermelo-Fraenkel set theory if large cardinals exist, it is also independent from these axioms. Gödel had demonstrated that if Zermelo-Fraenkel set theory is consistent, then so is the Continuum Hypothesis. Cohen took this further by proving that there are models where it does not hold, effectively establishing that both hypotheses coexist within different frameworks.
• Evaluate how Paul Cohen's contributions influence current research directions in set theory.
  • Cohen's innovative techniques and insights have profoundly influenced contemporary research in set theory by providing tools and methodologies for examining independence results. His work laid the groundwork for exploring new axioms and large cardinal theories, motivating researchers to investigate deeper questions about infinity and its implications. The ongoing study of forcing and related concepts continues to drive advancements in foundational mathematics, ensuring Cohen's legacy remains central to modern exploration in this field.

Related terms

Continuum Hypothesis: A hypothesis in set theory that posits there are no sets with cardinality strictly between that of the integers and the real numbers.

Forcing:

A technique developed by Cohen that allows mathematicians to construct models of set theory in which certain propositions can be shown to be true or false, thereby demonstrating their independence.

Gödel's Constructible Universe: A model of set theory introduced by Kurt Gödel where all sets are 'constructible', providing a framework in which the Continuum Hypothesis can be shown to be consistent with Zermelo-Fraenkel set theory.