Cohen's forcing technique

Definition

Cohen's forcing technique is a method used in set theory to construct models of set theory in which certain propositions can be shown to be true or false. This powerful technique allows mathematicians to add new sets to a model without changing the properties of the original model, thus enabling the exploration of independence results, particularly regarding the Axiom of Choice and various cardinalities. It revolutionized the way mathematicians approach questions about the consistency and independence of mathematical statements.

5 Must Know Facts For Your Next Test

1. Cohen's forcing technique was introduced by Paul Cohen in the 1960s as a way to show that the Continuum Hypothesis is independent of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC).
2. The method involves extending a model of set theory by adding a generic filter, allowing for the construction of new sets while preserving the original structure.
3. Cohen's forcing can be applied to prove that certain statements, such as 'There is no set whose cardinality is strictly between that of the integers and the real numbers,' are independent of ZFC.
4. The technique has applications beyond independence results; it also helps in understanding the structure and behavior of models in set theory.
5. Forcing has become a fundamental tool in modern set theory, influencing various areas including topology, combinatorial set theory, and the study of large cardinals.

Review Questions

• How does Cohen's forcing technique allow mathematicians to explore the independence of propositions related to set theory?
  • Cohen's forcing technique provides a systematic way to construct models where certain propositions can be made true or false without altering the foundational properties of the original model. By adding new sets through a process involving generic filters, mathematicians can demonstrate that specific statements, such as the Continuum Hypothesis, do not follow from standard axioms like ZFC. This ability to manipulate models makes it possible to establish independence results and understand deeper implications within set theory.
• In what ways has Cohen's forcing technique influenced modern mathematical logic and set theory?
  • Cohen's forcing technique has fundamentally transformed how mathematicians approach problems in set theory by providing tools for proving independence results. It has led to significant discoveries regarding the consistency of various mathematical statements and has opened pathways to further research into large cardinals and their properties. Additionally, forcing is instrumental in exploring more complex structures within models, impacting fields like topology and combinatorial set theory.
• Evaluate the implications of Cohen's forcing on the Axiom of Choice and how it relates to the broader landscape of mathematical theories.
  • Cohen's forcing has profound implications for the Axiom of Choice by demonstrating that this principle does not hold universally across all models of set theory. Through specific constructions using forcing, mathematicians can show scenarios where choices lead to contradictions or inconsistencies. This evaluation not only deepens our understanding of choice-related propositions but also situates the Axiom within a larger context where its acceptance can vary depending on the foundational framework employed, thus reshaping our views on mathematical certainty.

Related terms

Set Theory: A branch of mathematical logic that studies sets, which are collections of objects, and the relationships between them.

Axiom of Choice:

A principle in set theory stating that given a collection of non-empty sets, it is possible to choose one element from each set.

Independence Results: Results that demonstrate certain propositions cannot be proven or disproven using a given set of axioms, implying they are independent of those axioms.