Measurable cardinals

Definition

Measurable cardinals are a special type of large cardinal, which is a cardinal number that is so large that it cannot be reached by any standard set-theoretic operations. They have the property of being 'measurable', meaning there exists a non-trivial elementary embedding from the cardinal into itself, which preserves the structure of sets. This property connects them deeply with concepts like the continuum hypothesis and the consistency of various set-theoretic statements, making them crucial in contemporary research within the field.

5 Must Know Facts For Your Next Test

1. Measurable cardinals are uncountable and possess a specific property called 'ultrafilter', which allows for the construction of non-trivial elementary embeddings.
2. The existence of a measurable cardinal implies that certain statements in set theory, like the failure of the continuum hypothesis, can be established.
3. They are significantly larger than other large cardinals, making their study pivotal for understanding the hierarchy of infinities.
4. Measurable cardinals play a crucial role in determining the consistency of various mathematical propositions within set theory.
5. The study of measurable cardinals leads to deep implications regarding the nature of infinity and how different sizes of infinity relate to each other.

Review Questions

• How do measurable cardinals relate to large cardinals and their significance in set theory?
  • Measurable cardinals are a subset of large cardinals, which represent some of the largest types of cardinal numbers in set theory. Their unique property of having non-trivial elementary embeddings gives them significant strength and allows researchers to derive many important results regarding consistency and independence. Understanding measurable cardinals is essential for grasping how different sizes of infinity operate and interact within the broader framework of set theory.
• Discuss the implications of measurable cardinals on the continuum hypothesis and how they influence modern research directions in set theory.
  • The existence of measurable cardinals has profound implications on the continuum hypothesis, as their presence can lead to models of set theory where the hypothesis fails. This is because measurable cardinals provide a framework for creating sets that exceed standard cardinalities. Researchers are exploring these connections to better understand not only the structure of infinite sets but also how these properties affect foundational questions in mathematics, making measurable cardinals a focus of contemporary investigations.
• Evaluate the role of forcing in proving results related to measurable cardinals and its impact on our understanding of set theory.
  • Forcing is a powerful method used to establish consistency results in set theory, particularly concerning large cardinals like measurable cardinals. By using forcing, mathematicians can construct models in which certain properties hold or fail, allowing them to explore the boundaries of what can be proven about measurable cardinals. This has significant implications for our understanding of set theory as it enables researchers to manipulate notions of size and infinity in ways that reveal deeper truths about mathematical structures, illustrating how these concepts are interlinked.

Related terms

Large Cardinals: A class of cardinals that extend the standard concept of cardinality, often having strong properties and implications for set theory.

Elementary Embedding: A function between two structures that preserves their properties and relations, important for understanding measurable cardinals.

Forcing: A technique used in set theory to prove consistency and independence results, including those involving large cardinals and the continuum hypothesis.