libterm.com A strongly compact cardinal is a type of large cardinal in set theory characterized by its powerful compactness properties, extending those of measurable cardinals. These cardinals have significant implications for the structure of the set-theoretic universe, influencing principles like the existence of ultrafilters and impacting model theory. Explore the rest of the article to understand how strongly compact cardinals shape advanced mathematical logic and your comprehension of infinite sets.
Table of Comparison
| Feature | Strongly Compact Cardinal | Measurable Cardinal |
|---|---|---|
| Definition | Cardinal k such that every k-complete filter can be extended to a k-complete ultrafilter | Cardinal k with a non-principal, k-additive, 0-1 valued measure (k-complete ultrafilter) |
| Consistency Strength | Stronger than measurable cardinals | Weaker than strongly compact cardinals |
| Filter Extension Property | Every k-complete filter extends to a k-complete ultrafilter | Exists a k-complete, non-principal ultrafilter (measure) |
| Compactness | Implies k-compactness for infinitary logic (L_{k, k}) | No direct compactness equivalence |
| Examples | Stronger large cardinal, often above measurable | First large cardinal notion beyond inaccessible |
| Relationship | Every strongly compact cardinal is measurable | Not every measurable cardinal is strongly compact |
Introduction to Large Cardinals
Strongly compact cardinals generalize measurable cardinals by requiring the existence of k-complete ultrafilters that extend beyond measures to support strong compactness properties in infinitary logic. Measurable cardinals are characterized by the presence of a k-additive, non-principal ultrafilter, which enables the construction of elementary embeddings and reflects large cardinal strength. Both cardinals reside at higher levels in the large cardinal hierarchy, with strongly compact cardinals exhibiting stronger combinatorial and reflection properties than measurable cardinals.
Definition of Strongly Compact Cardinals
Strongly compact cardinals are large cardinals characterized by the property that every l-complete filter can be extended to a l-complete ultrafilter for all cardinals l >= k, where k is the strongly compact cardinal. This definition implies that strongly compact cardinals exhibit a higher degree of compactness and are strictly stronger than measurable cardinals, which are defined by the existence of a non-principal k-complete ultrafilter over k. The strong compactness property leads to profound consequences in model theory and set theory, particularly in the analysis of infinitary logic and the structure of large cardinal hierarchies.
Definition of Measurable Cardinals
Measurable cardinals are defined as uncountable cardinals k for which there exists a non-trivial k-additive, 0-1 valued measure defined on all subsets of k, reflecting a high degree of largeness and allowing the construction of ultrapowers. Strongly compact cardinals extend this notion by requiring every k-complete filter to be extendable to a k-complete ultrafilter, implying stronger combinatorial properties and compactness principles in infinitary logic. The existence of a measurable cardinal guarantees a strong large cardinal structure, while strongly compact cardinals entail even more powerful consistency strength and structural consequences in set theory.
Hierarchy of Large Cardinals
Strongly compact cardinals are large cardinals that extend measurable cardinals by requiring every kappa-complete filter to be extendable to a kappa-complete ultrafilter, reflecting a higher level of compactness in infinitary logic. Measurable cardinals possess a non-trivial, kappa-complete ultrafilter, marking the starting point of large cardinal strength beyond inaccessible cardinals. In the hierarchy of large cardinals, strongly compact cardinals are strictly stronger than measurable cardinals, implying stronger consistency strength and more extensive combinatorial properties.
Key Properties of Strongly Compact Cardinals
Strongly compact cardinals extend the concept of measurable cardinals by satisfying a stronger filter extension property that applies to all k-complete ultrafilters, ensuring the existence of fine, k-complete ultrafilters on every set of size at least k. Key properties of strongly compact cardinals include their ability to reflect large cardinal characteristics down to smaller cardinals and their influence on compactness in infinitary logics like Lk,l. Unlike measurable cardinals, which guarantee only a k-complete non-principal ultrafilter on k, strongly compact cardinals induce a robust form of compactness that impacts combinatorial set theory and model theory in a fundamental way.
Key Properties of Measurable Cardinals
Measurable cardinals are large cardinals characterized by the existence of a non-principal, k-complete ultrafilter, enabling the construction of a non-trivial elementary embedding from the universe V into a transitive class M. They possess strong completeness properties, allowing for the definition of a measure that extends the notion of "almost everywhere" in set theory. Compared to strongly compact cardinals, measurable cardinals ensure the existence of a fine, k-additive measure, making them a foundational concept in large cardinal hierarchy and inner model theory.
Major Differences Between Strongly Compact and Measurable Cardinals
Strongly compact cardinals exhibit a stronger level of compactness by requiring every k-complete filter to extend to a k-complete ultrafilter, surpassing the measurable cardinal's definition based on the existence of a k-additive non-principal ultrafilter. Measurable cardinals ensure the existence of a single k-complete non-principal ultrafilter, while strongly compact cardinals demand compactness properties that affect entire logical languages and lead to stronger large cardinal axioms. The major difference lies in their hierarchy and structural impact: every strongly compact cardinal is measurable, but not every measurable cardinal is strongly compact, reflecting a strict increase in large cardinal strength and consequences for combinatorial set theory.
Interrelations and Implications
Strongly compact cardinals extend the concept of measurable cardinals by requiring every k-complete filter to be extendable to a k-complete ultrafilter, implying measurability. The existence of a strongly compact cardinal typically results in the existence of measurable cardinals concentrated below it, reflecting a hierarchical relationship. Interrelations reveal that strong compactness implies more robust large cardinal properties and has profound implications for model theory and set-theoretic topology.
Examples and Constructions
Strongly compact cardinals extend the concept of measurable cardinals by supporting finer compactness properties, with every strongly compact cardinal also being measurable but not vice versa. Classical constructions of measurable cardinals involve ultrafilters and elementary embeddings derived from large cardinal axioms in set theory, while strongly compact cardinals require more elaborate models ensuring the compactness of infinitary logics like L_{k,l}. Examples include the first measurable cardinal k admitting a k-complete non-principal ultrafilter, whereas strongly compact cardinals guarantee compactness principles for all l >= k, exemplified by the extension of the filter-based approach to infinitary compactness results.
Open Problems and Research Directions
Strongly compact cardinals extend the concept of measurable cardinals by requiring every l-complete filter to extend to a l-complete ultrafilter, posing open problems related to the exact consistency strength and their behavior under forcing. Research continues to explore the precise boundaries between the hierarchy of large cardinals, particularly investigating the interactions between strong compactness, supercompactness, and reflection properties. Current directions focus on determining the relative consistency of strongly compact cardinals without measurable cardinals and understanding their implications for inner model theory and descriptive set theory.
Strongly compact cardinal Infographic
