9.2 Large Cardinals

Large cardinal theory studies strong axioms asserting the existence of infinite cardinals with properties far beyond those provable in ZFC, and these axioms are important because they measure the strength of set theoretic principles and provide a systematic hierarchy of stronger and stronger extensions of ordinary set theory.

The basic idea is that some cardinals are so large that the universe below them resembles the whole universe in certain structural ways, and this resemblance can be expressed through closure properties, reflection principles, measures, or elementary embeddings.

Definition 9.23 (Cardinal)

A cardinal is an ordinal used to measure the size of a set, where two sets have the same cardinality if there exists a bijection between them.

If $A$ and $B$ are sets, then:

means that there exists a bijection:

A cardinal $\kappa$ is infinite if it is not finite, and the first infinite cardinal is:

which is the cardinality of the natural numbers.

Definition 9.24 (Regular Cardinal)

An infinite cardinal $\kappa$ is regular if it cannot be written as the union of fewer than $\kappa$ many sets, each of size less than $\kappa$.

Equivalently, $\kappa$ is regular if:

where $\mathrm{cf}(\kappa)$ denotes the cofinality of $\kappa$.

The cofinality of $\kappa$ is the least cardinality of an unbounded subset of $\kappa$.

Example 9.25

The cardinal:

is regular, because no finite increasing sequence of natural numbers is cofinal in $\omega$.

The cardinal:

is singular, because:

Thus:

This means that $\aleph_\omega$ can be approached by a countable increasing sequence of smaller cardinals.

Definition 9.26 (Strong Limit Cardinal)

An infinite cardinal $\kappa$ is a strong limit cardinal if:

for every cardinal $\lambda < \kappa$.

This says that the power set operation applied below $\kappa$ never reaches $\kappa$.

The strong limit condition is a closure condition, because it says that many set forming operations below $\kappa$ remain below $\kappa$.

Definition 9.27 (Inaccessible Cardinal)

An uncountable cardinal $\kappa$ is inaccessible if it is regular and strong limit.

Thus $\kappa$ is inaccessible if:

1. $\kappa$ is uncountable.
2. $\mathrm{cf}(\kappa)=\kappa$.
3. For every $\lambda<\kappa$: $$2^\lambda < \kappa$$

An inaccessible cardinal is “inaccessible” because it cannot be reached from smaller cardinals by ordinary cardinal operations such as taking suprema of short sequences or taking power sets below it.

Lemma 9.28

If $\kappa$ is inaccessible, then $V_\kappa$ is a model of many axioms of ZFC.

Here:

denotes the $\kappa$ th level of the cumulative hierarchy.

Proof

The cumulative hierarchy is defined by:

and for limit ordinals $\lambda$:

Because $\kappa$ is a limit ordinal, $V_\kappa$ is closed under the earlier stages of the hierarchy.

The strong limit property ensures that power sets of sets of rank below $\kappa$ still have rank below $\kappa$ in the relevant sense, so the power set operation does not push us out of $V_\kappa$.

The regularity of $\kappa$ ensures that unions indexed by sets of size less than $\kappa$ remain below $\kappa$, so replacement and union type constructions do not escape $V_\kappa$.

Thus the closure properties of $\kappa$ make $V_\kappa$ behave like a small universe of sets.

The exact fragment of ZFC satisfied by $V_\kappa$ depends on the precise formulation, but the main point is that inaccessibility gives enough closure for $V_\kappa$ to satisfy the ordinary set constructions used throughout mathematics.

Definition 9.29 (Weakly Inaccessible Cardinal)

An uncountable cardinal $\kappa$ is weakly inaccessible if it is regular and a limit cardinal.

This is weaker than being inaccessible, because it does not require the strong limit condition.

Thus every inaccessible cardinal is weakly inaccessible, but the converse need not hold.

Definition 9.30 (Club Set)

Let $\kappa$ be an uncountable regular cardinal. A set $C \subseteq \kappa$ is closed unbounded, or club, if it satisfies the following two conditions.

First, $C$ is unbounded in $\kappa$, meaning that for every $\alpha<\kappa$ there exists $\beta \in C$ such that:

Second, $C$ is closed, meaning that whenever:

is an increasing sequence from $C$ of length $\lambda<\kappa$, and:

then:

Club sets represent large subsets of a regular cardinal.

Definition 9.31 (Stationary Set)

Let $\kappa$ be an uncountable regular cardinal. A set $S \subseteq \kappa$ is stationary if it meets every club subset of $\kappa$.

That is, for every club set $C \subseteq \kappa$:

Stationary sets are large in a strong combinatorial sense, because they cannot be avoided by closed unbounded sets.

Definition 9.32 (Mahlo Cardinal)

An inaccessible cardinal $\kappa$ is Mahlo if the set of inaccessible cardinals below $\kappa$ is stationary in $\kappa$.

In symbols, $\kappa$ is Mahlo if:

is stationary in $\kappa$.

A Mahlo cardinal reflects inaccessibility many times below itself, because inaccessible cardinals occur throughout $\kappa$ in a stationary way.

Lemma 9.33

Every Mahlo cardinal is inaccessible.

Proof

This follows directly from the definition, since a Mahlo cardinal is defined to be an inaccessible cardinal with an additional stationary reflection property.

The Mahlo condition strengthens inaccessibility by requiring not only that $\kappa$ itself has strong closure properties, but also that many smaller cardinals below $\kappa$ have those same closure properties.

Definition 9.34 (Ultrafilter)

Let $X$ be a set. An ultrafilter on $X$ is a collection $U \subseteq \mathcal{P}(X)$ satisfying the following conditions.

First:

and:

Second, if $A \in U$ and $A \subseteq B \subseteq X$, then:

Third, if $A,B \in U$, then:

Fourth, for every $A \subseteq X$, exactly one of the following holds:

or:

An ultrafilter chooses, for every subset of $X$, whether that subset is large or its complement is large.

Definition 9.35 (Principal and Nonprincipal Ultrafilters)

An ultrafilter $U$ on $X$ is principal if there exists $x \in X$ such that:

An ultrafilter is nonprincipal if it is not principal.

A principal ultrafilter concentrates on a single point, while a nonprincipal ultrafilter measures largeness in a way that does not reduce to membership of one fixed element.

Definition 9.36 ($\kappa$ Complete Ultrafilter)

Let $\kappa$ be an infinite cardinal. An ultrafilter $U$ is $\kappa$ complete if whenever:

is a collection of members of $U$ with:

then:

Thus $\kappa$ completeness says that the ultrafilter is closed under intersections of fewer than $\kappa$ many large sets.

Definition 9.37 (Measurable Cardinal)

An uncountable cardinal $\kappa$ is measurable if there exists a nonprincipal $\kappa$ complete ultrafilter on $\kappa$.

Such an ultrafilter is called a measure on $\kappa$.

The word “measure” is used because the ultrafilter behaves like a two valued measure, where a subset of $\kappa$ has measure one if it belongs to $U$, and measure zero if its complement belongs to $U$.

Lemma 9.38

If $\kappa$ is measurable, then $\kappa$ is regular.

Proof

Let $U$ be a nonprincipal $\kappa$ complete ultrafilter on $\kappa$.

Suppose, toward a contradiction, that $\kappa$ is singular. Then there is a cofinal sequence:

with:

and:

Define:

viewed as the set of ordinals below $\kappa_i$.

Since the sequence is cofinal, we have:

Because $U$ is an ultrafilter and $\kappa \in U$, at least one part of this cover must be large in a sense compatible with $\kappa$ completeness.

More directly, if every $A_i$ were not in $U$, then each complement:

would belong to $U$.

Since $\lambda<\kappa$ and $U$ is $\kappa$ complete:

But:

This contradicts:

Therefore some $A_i$ belongs to $U$.

But $A_i$ has size less than $\kappa$, and a nonprincipal $\kappa$ complete ultrafilter on $\kappa$ cannot contain a bounded subset of $\kappa$, because such a set is the union of fewer than $\kappa$ many singletons, and no singleton can belong to a nonprincipal ultrafilter.

This contradiction shows that $\kappa$ is regular.

Lemma 9.39

If $\kappa$ is measurable, then $\kappa$ is inaccessible.

Proof

By Lemma 9.38, $\kappa$ is regular.

It remains to explain why $\kappa$ is a strong limit in the usual large cardinal hierarchy context.

A standard theorem shows that every measurable cardinal is strongly inaccessible, and the proof uses the existence of a $\kappa$ complete nonprincipal ultrafilter to derive strong closure and reflection properties below $\kappa$.

The essential reason is that a measure on $\kappa$ gives enough coherence to form an ultrapower of the universe, producing an elementary embedding:

with critical point:

The existence of such an embedding implies strong restrictions on the structure below $\kappa$, including regularity and strong limit behavior.

Thus measurable cardinals are far stronger than inaccessible cardinals.

Definition 9.40 (Elementary Embedding)

Let $M$ and $N$ be structures in the same language. A function:

is an elementary embedding if for every formula $\varphi(x_1,\dots,x_n)$ and all parameters $a_1,\dots,a_n\in M$:

if and only if:

Elementary embeddings preserve the truth of all first order statements.

Definition 9.41 (Critical Point)

Let:

be a nontrivial elementary embedding. The critical point of $j$ is the least ordinal $\alpha$ such that:

It is denoted:

The critical point is the first place where the embedding moves the universe.

Theorem 9.42 (Measurability and Elementary Embeddings)

A cardinal $\kappa$ is measurable if and only if there exists a transitive class $M$ and a nontrivial elementary embedding:

such that:

Proof

We give the main construction in the direction from a measure to an embedding.

Assume $\kappa$ is measurable, and let $U$ be a nonprincipal $\kappa$ complete ultrafilter on $\kappa$.

Define an equivalence relation on functions:

by:

if and only if:

Let:

denote the equivalence class of $f$.

The ultrapower consists of these equivalence classes:

Membership in the ultrapower is defined by:

if and only if:

The map:

is defined by sending each set $x$ to the equivalence class of the constant function with value $x$:

Los theorem for ultrapowers shows that $j$ is elementary.

Because $U$ is nonprincipal, all ordinals below $\kappa$ are fixed by $j$, while $\kappa$ itself is moved.

Hence:

The converse direction starts with an elementary embedding $j:V\to M$ with critical point $\kappa$ and defines:

One verifies that $U$ is a nonprincipal $\kappa$ complete ultrafilter on $\kappa$.

Thus measurability is equivalent to the existence of such an elementary embedding.

Definition 9.43 (Strong Cardinal)

A cardinal $\kappa$ is strong if for every ordinal $\theta$ there exists an elementary embedding:

with:

such that:

This means that $\kappa$ can see arbitrarily large initial segments of the universe through elementary embeddings.

Definition 9.44 (Supercompact Cardinal)

A cardinal $\kappa$ is supercompact if for every cardinal $\lambda \geq \kappa$ there exists an elementary embedding:

with:

and:

such that $M$ is closed under $\lambda$ sequences:

Supercompactness is much stronger than measurability, because it asserts the existence of elementary embeddings with very high closure properties for every larger cardinal $\lambda$.

Definition 9.45 (Woodin Cardinal)

A cardinal $\delta$ is Woodin if for every function:

there exists a cardinal $\kappa<\delta$ and an elementary embedding:

with:

such that:

and:

Woodin cardinals are central in modern descriptive set theory and inner model theory, especially in the study of determinacy and projective sets of reals.

Large Cardinal Hierarchy

Large cardinals form a hierarchy ordered by consistency strength. A simplified part of the hierarchy is:

The symbol $<$ here should be read informally as “has lower consistency strength than”, not as an order relation between cardinals.

A stronger large cardinal axiom usually implies the consistency of weaker large cardinal axioms, provided the surrounding metatheory is strong enough.

Definition 9.46 (Consistency Strength)

Let $T_1$ and $T_2$ be theories. We say that $T_2$ has at least the consistency strength of $T_1$ if:

If $T_2$ proves the consistency of $T_1$, then $T_2$ is stronger than $T_1$ in the sense of proof theoretic or consistency strength.

Large cardinal axioms are often compared by this notion.

Example 9.47

The theory:

has greater consistency strength than ZFC, because if $\kappa$ is inaccessible, then $V_\kappa$ satisfies a large part of ZFC and in many standard formulations gives a model witnessing the consistency of ZFC.

Similarly:

has greater consistency strength than:

because measurability implies much stronger structural properties than inaccessibility.

Why Large Cardinals Matter

Large cardinals matter because they provide a calibrated scale for measuring the strength of mathematical statements that cannot be settled by ordinary ZFC alone.

In independence results, one often proves that a statement follows from a certain large cardinal axiom, or that its consistency is equivalent to the consistency of some large cardinal hypothesis.

Thus large cardinals play a role similar to measuring instruments: they do not merely add new sets, but also organize the relative strength of theories and principles across set theory, model theory, descriptive set theory, and the foundations of mathematics.