Measurable cardinals

Definition

Measurable cardinals are a special type of large cardinal number that possess a non-trivial elementary embedding from the cardinal into itself. This property implies that they are uncountable and have a rich structure, allowing for the construction of a non-principal ultrafilter over the set. These cardinals have significant implications for both set theory and the foundations of mathematics, particularly in understanding the hierarchy of infinite sets and the relationships between different axioms of set theory.

5 Must Know Facts For Your Next Test

1. Measurable cardinals are always greater than any countable cardinal, meaning they have a size larger than any set that can be put into one-to-one correspondence with the natural numbers.
2. The existence of a measurable cardinal implies that ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) cannot prove its own consistency, thus highlighting a critical aspect of incompleteness.
3. Measurable cardinals exhibit properties related to elementary embeddings, which means they can reflect certain logical truths back into themselves.
4. The existence of measurable cardinals leads to strong consequences in forcing and the study of independence results in set theory, such as those concerning the Continuum Hypothesis.
5. If a measurable cardinal exists, then it can be shown that there is an inner model containing all ordinals and having certain desirable properties related to large cardinals.

Review Questions

• How do measurable cardinals relate to the concept of large cardinals and what makes them distinct?
  • Measurable cardinals are a specific type of large cardinal characterized by their ability to support non-trivial elementary embeddings. While all measurable cardinals are large cardinals due to their strength and implications, not all large cardinals are measurable. The uniqueness of measurable cardinals lies in their capacity to define non-principal ultrafilters and their connection to the foundations of mathematics, offering deeper insights into set theory's hierarchy.
• Discuss the implications of measurable cardinals on ZFC set theory and its consistency.
  • The existence of measurable cardinals has profound implications for ZFC set theory as it suggests that ZFC cannot prove its own consistency. This arises from the fact that if a measurable cardinal exists, it allows for models where certain statements can be true or false independent of ZFC axioms. Thus, measurable cardinals serve as a benchmark for understanding limitations within set theory and contribute to discussions on the foundations and consistency proofs in mathematics.
• Evaluate how measurable cardinals influence our understanding of independence results within set theory.
  • Measurable cardinals greatly enhance our comprehension of independence results in set theory by providing examples where certain propositions cannot be proven or disproven within ZFC. Their existence leads to insights about other significant hypotheses, such as the Continuum Hypothesis, which may hold true or false depending on whether large cardinal axioms are adopted. The study of these cardinals not only illuminates relationships between different set theoretic propositions but also informs us about potential extensions and limitations of mathematical logic itself.

Related terms

large cardinals: A class of cardinals that extend the standard notion of infinity in set theory, each exhibiting strong properties and implications for the consistency of various mathematical theories.

elementary embedding: A function between two structures that preserves the truth of formulas, crucial in the study of measurable cardinals as it allows for their defining properties.

ultrafilter: A collection of subsets of a set that extends the notion of filters, particularly important in the context of measurable cardinals where non-principal ultrafilters play a key role.