Cohen's forcing technique

Definition

Cohen's forcing technique is a method developed by Paul Cohen to show the independence of certain mathematical statements from Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). It allows mathematicians to construct models of set theory in which specific propositions can be true or false, illustrating that some questions, like the Continuum Hypothesis, cannot be resolved within standard axiomatic frameworks. This technique is crucial for understanding how certain mathematical truths can exist in multiple contexts.

5 Must Know Facts For Your Next Test

1. Cohen's forcing technique was first introduced in the 1960s and became instrumental in proving the independence of the Continuum Hypothesis from ZFC.
2. The method involves adding new sets to a model of set theory, which alters its properties and can make certain propositions hold or fail.
3. Cohen showed that it is possible to create models in which both the Continuum Hypothesis is true and models in which it is false, demonstrating its independence.
4. Forcing relies on the concept of a partially ordered set (poset) that determines how new sets are constructed within a model.
5. The technique has far-reaching implications, not only in set theory but also in areas like topology, algebra, and the philosophy of mathematics.

Review Questions

• How does Cohen's forcing technique illustrate the concept of independence in mathematical logic?
  • Cohen's forcing technique demonstrates independence by constructing models of set theory where specific statements, like the Continuum Hypothesis, can be shown to be both true and false depending on the model used. By adding new sets through forcing, Cohen illustrated that some mathematical questions cannot be definitively resolved within Zermelo-Fraenkel set theory with the Axiom of Choice. This shows that independence is not just a theoretical idea but can be practically achieved through careful construction.
• Discuss the significance of Cohen's forcing technique in relation to other independence results in set theory.
  • Cohen's forcing technique holds significant importance as it was one of the first methods used to prove independence results such as the Continuum Hypothesis and later contributed to understanding other propositions like Martin's Axiom. This approach paved the way for further exploration into independent statements and shaped our understanding of what can be proved within various axiomatic systems. The implications extend beyond these results, influencing how mathematicians approach questions about truth and provability in set theory.
• Evaluate how Cohen's forcing technique reshapes our understanding of mathematical truths in different models and its broader implications.
  • Cohen's forcing technique reshapes our understanding by showing that mathematical truths are not absolute but can depend on the chosen axioms and models. By demonstrating that certain statements can exist in various forms across different models, it emphasizes the complexity and richness of set theory and logic. This has broader implications for philosophy, particularly regarding what constitutes truth in mathematics, challenging traditional views about certainty and proof, and encouraging a more nuanced perspective on mathematical foundations.

Related terms

Independence:

A property of a statement in mathematics where it cannot be proven true or false using a given set of axioms.

Zermelo-Fraenkel Set Theory (ZF): A foundational system for mathematics that provides a formal structure for set theory without assuming the Axiom of Choice.

Continuum Hypothesis:

A conjecture about the possible sizes of infinite sets, stating there is no set whose size is strictly between that of the integers and the real numbers.