Inaccessible cardinals

Definition

Inaccessible cardinals are a type of large cardinal that cannot be reached by certain set-theoretic operations, such as taking power sets or forming unions of fewer than their own size. They are significant because they serve as a boundary for the consistency of various mathematical statements, including the Continuum Hypothesis. These cardinals are often used in advanced set theory to investigate the nature of infinity and the foundations of mathematics.

5 Must Know Facts For Your Next Test

1. Inaccessible cardinals are not provably existent in Zermelo-Fraenkel set theory (ZF) alone but are consistent with ZF if large cardinals exist.
2. These cardinals are uncountable and can be thought of as 'large' because any cardinal smaller than an inaccessible cardinal cannot reach it through standard operations.
3. The existence of inaccessible cardinals implies the existence of many other large cardinals and can lead to interesting implications in model theory.
4. Inaccessible cardinals have a closure property under certain operations, which means that if you have an inaccessible cardinal, taking power sets or unions below its size will not yield an inaccessible cardinal.
5. When discussing forcing, inaccessible cardinals can help establish models in which the properties of CH hold or fail, showing their importance in set-theoretic independence results.

Review Questions

• What role do inaccessible cardinals play in understanding the limitations and capabilities of set-theoretic operations?
  • Inaccessible cardinals highlight the limitations of set-theoretic operations since they cannot be reached by taking power sets or forming unions of fewer than their size. This characteristic allows mathematicians to better understand the structure and hierarchy within set theory, as they serve as a threshold that separates smaller cardinals from those that have properties reflective of larger infinities. Essentially, they help map out the landscape of infinite sizes and how different levels interact through various operations.
• Discuss how the existence of inaccessible cardinals relates to the Continuum Hypothesis and its implications in set theory.
  • The existence of inaccessible cardinals has profound implications for the Continuum Hypothesis (CH). Since inaccessible cardinals provide a framework within which many models can be constructed, they help mathematicians analyze whether CH holds or fails in those models. When large cardinals like inaccessible ones exist, it allows researchers to show that certain propositions regarding the continuum can be independent from ZF. This interaction emphasizes how large cardinals influence foundational aspects in set theory.
• Evaluate how forcing techniques utilize inaccessible cardinals to address independence results in set theory.
  • Forcing techniques leverage the properties of inaccessible cardinals to create models that reveal independence results in set theory. By using these large cardinals, mathematicians can establish conditions under which CH holds or fails within various models, thus demonstrating that certain propositions cannot be proven or disproven using standard axioms alone. This powerful method shows how inaccessible cardinals can guide researchers through complex discussions surrounding the foundational aspects of mathematics, pushing forward our understanding of what can be known about infinite structures.

Related terms

large cardinals: Large cardinals are a class of infinite cardinal numbers that possess strong properties, often used to prove the consistency of other mathematical statements.

Continuum Hypothesis (CH): The Continuum Hypothesis is a statement about the size of the set of real numbers, asserting that there are no sets whose size is strictly between that of the integers and the real numbers.

forcing: Forcing is a method used in set theory to construct new models of set theory, often employed to show the independence of certain propositions from Zermelo-Fraenkel set theory with the Axiom of Choice.