In set theory, a strong cardinal is a type of large cardinal, named for its significant "strength" in terms of consistency and the existence of elementary embeddings. Strong cardinals are among the most powerful large cardinal hypotheses studied, implying the existence of many weaker large cardinals.
Formal Definition
A cardinal $\kappa$ is said to be strong if for every ordinal $\alpha$, there exists an elementary embedding $j: V \to M$ such that:- $\text{crit}(j) = \kappa$
- $V_\alpha \subseteq M$
Here:
- $V$ denotes the Von Neumann universe (the universe of all sets).
- $M$ is a transitive inner model of ZFC.
- $j$ is an elementary embedding, meaning it preserves truth of first-order formulas.
- $\text{crit}(j)$ is the critical point of $j$, which is the least ordinal $\delta$ such that $j(\delta) eq \delta$. For a strong cardinal, this critical point is the cardinal itself, $\kappa$.
- $V_\alpha$ represents the $\alpha$-th stage of the Von Neumann hierarchy, which contains all sets whose rank is less than $\alpha$. The condition $V_\alpha \subseteq M$ means that $M$ contains the entire initial segment $V_\alpha$ of the universe $V$. This implies that $j(\kappa) > \alpha$.
Properties and Implications
The existence of a strong cardinal has profound implications within set theory:- Consistency: The existence of a strong cardinal cannot be proven within ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice), assuming ZFC is consistent. If ZFC + "there exists a strong cardinal" is consistent, then ZFC itself is consistent.
- Measurability: Every strong cardinal is a measurable cardinal. This means it carries a non-trivial $\kappa$-complete ultrafilter.
- Supercompactness: Every strong cardinal is supercompact. This means that for every $\lambda \ge \kappa$, there exists a fine, normal, $\kappa$-complete ultrafilter on $P_\kappa(\lambda)$. The definition of strong cardinals given above is indeed stronger than the typical definition of supercompactness, implying it.
- Limit of Large Cardinals: If $\kappa$ is a strong cardinal, then $\kappa$ must be a limit of strong cardinals. It is also a limit of measurable, supercompact, and other large cardinals below it.
- Higher Consistency Strength: Strong cardinals occupy a high position in the large cardinal hierarchy, implying the existence of many other large cardinals further down the hierarchy, such as measurable, supercompact, Vopěnka cardinals (under some definitions), and others.
Relationship to Other Large Cardinals
Strong cardinals are part of the large cardinal hierarchy. Their existence implies the existence of many "weaker" large cardinals. The standard hierarchy includes:- Strong cardinals
- Supercompact cardinals
- Measurable cardinals
- Inaccessible cardinals
- Weakly compact cardinals
The existence of a strong cardinal is a very powerful axiom of infinity, often used to establish the consistency of various extensions of ZFC or to prove results about determinacy axioms.
See Also
- Measurable cardinal
- Supercompact cardinal
- Elementary embedding
- Large cardinal
- Von Neumann universe