8.5 Independence Phenomena

An independence result shows that a given statement cannot be proved or refuted from a given system of axioms, assuming that the system itself is consistent. In set theory, many natural and important statements turn out to be independent of the standard axioms.

The study of independence phenomena reveals that the axioms of ZF and ZFC do not determine a single complete picture of the universe of sets. Instead, different models of set theory may satisfy different additional principles.

Definition 8.84 (Independence)

Let $T$ be a theory and let $\varphi$ be a sentence. The sentence $\varphi$ is independent of $T$ if:

1. $T \nvdash \varphi$.

2. $T \nvdash ¬\varphi$.

In practice, independence is established by proving two relative consistency statements:

and:

When both are known, the theory $T$ cannot decide $\varphi$.

The Axiom of Choice

The axiom of choice is independent of ZF. One direction is given by the constructible universe.

Theorem 8.85

If ZF is consistent, then:

is consistent.

Proof

By the constructibility theorem, if ZF has a model, then the constructible universe $L$ inside that model satisfies ZF and also satisfies the axiom of choice. Therefore:

Theorem 8.86

If ZF is consistent, then:

is consistent.

Proof

The proof uses permutation models, also called Fraenkel Mostowski models. The idea is to start with a model of ZF that contains a set of atoms and then consider a class of sets that are invariant under a group of permutations of the atoms.

One restricts attention to sets that are symmetric with respect to these permutations. In such a model, it can be arranged that there is no global choice function, because any attempt to define such a function would break the symmetry.

Thus one obtains a model of ZF in which the axiom of choice fails. Therefore:

Corollary 8.87

The axiom of choice is independent of ZF.

Proof

Theorem 8.85 shows that:

Theorem 8.86 shows that:

Therefore, if ZF is consistent, then neither AC nor its negation can be proved from ZF.

The Continuum Hypothesis

The continuum hypothesis is the statement:

It asserts that there is no cardinal strictly between the cardinality of the natural numbers and the cardinality of the real numbers.

This statement was one of the earliest central problems in set theory. It turns out to be independent of ZFC.

Theorem 8.88

If ZF is consistent, then:

is consistent.

Proof

By the constructibility theorem, if ZF is consistent, then:

is consistent.

Inside $L$, both the axiom of choice and the continuum hypothesis hold. Therefore:

Theorem 8.89

If ZFC is consistent, then:

is consistent.

Proof

The proof uses forcing, introduced by Paul Cohen. The idea is to extend a given model of set theory to a larger model in which new subsets of $\mathbb{N}$ are added in a controlled way.

One constructs a forcing notion $\mathbb{P}$ and considers generic filters $G$ over $\mathbb{P}$. The extension $M[G]$ of a model $M$ contains new sets that were not present in $M$.

By carefully choosing the forcing notion, one can ensure that the resulting model satisfies:

This shows that:

Corollary 8.90

The continuum hypothesis is independent of ZFC.

Proof

Theorem 8.88 shows that CH is consistent with ZFC, assuming ZF is consistent. Theorem 8.89 shows that the negation of CH is also consistent with ZFC, assuming ZFC is consistent.

Therefore CH is independent of ZFC.

The Method of Forcing

Forcing is the main method used to prove independence results in modern set theory.

The general idea is as follows.

Start with a model $M$ of ZFC. Construct a partially ordered set $\mathbb{P}$ inside $M$, called a forcing notion. Elements of $\mathbb{P}$ are interpreted as approximations to new objects that will be added to the model.

A filter $G \subseteq \mathbb{P}$ is called generic over $M$ if it meets every dense subset of $\mathbb{P}$ that belongs to $M$.

One then forms an extension:

This extension contains all elements of $M$ together with new objects determined by $G$.

The forcing construction is designed so that:

1. $M[G]$ satisfies ZFC.

2. $M[G]$ satisfies some additional property $\varphi$ that may not hold in $M$.

This gives a model of $\mathrm{ZFC}+\varphi$, and hence a relative consistency result.

Generic Extensions

The extension $M[G]$ is called a generic extension of $M$.

The key properties are:

1. Every element of $M$ remains in $M[G]$.

2. New elements are added in a controlled way.

3. The axioms of ZFC remain true in $M[G]$.

4. The construction preserves many structural properties of the original model.

The difficulty lies in constructing $G$ and proving that it has the required genericity properties.

Independence Beyond CH

The continuum hypothesis is only the first example of independence. Many other statements about infinite sets are independent of ZFC.

Examples include:

1. The generalized continuum hypothesis.

2. Statements about the structure of the real line.

3. Combinatorial principles such as Suslin’s hypothesis.

4. Statements about large cardinals.

Some of these statements require stronger assumptions to establish consistency results.

Large Cardinals and Strength

Large cardinal axioms assert the existence of infinite cardinals with strong combinatorial or structural properties.

These axioms are not provable in ZFC, but they are often consistent relative to even stronger axioms.

Independence results involving large cardinals often take the form:

This means that if very strong infinite objects exist, then certain additional statements are consistent.

Absoluteness and Robustness

Some statements are absolute between different models of set theory. An absolute statement has the same truth value in many different models.

Independence phenomena often arise for statements that are not absolute. Their truth can change when passing from one model to another.

For example, the statement that a set is finite is absolute, while statements about cardinalities of infinite sets are often not absolute.

Understanding which statements are absolute and which are not is an important part of modern set theory.

The Role of Independence

Independence results show that the axioms of set theory do not uniquely determine all mathematical truths about sets.

Instead, one may adopt additional axioms depending on the mathematical goals. Different choices lead to different but internally consistent theories.

This situation is similar to geometry, where Euclidean and non Euclidean geometries arise from different choices of axioms.

In set theory, independence results guide the search for new axioms that capture natural mathematical principles while extending the standard framework.

Summary of Main Results

The main independence phenomena discussed in this section are:

1. The axiom of choice is independent of ZF.

2. The continuum hypothesis is independent of ZFC.

3. These results are established by combining inner model methods, such as the constructible universe, with forcing and permutation models.

4. Independence shows that multiple consistent extensions of set theory exist, each with its own structure of sets.

These ideas form the foundation for modern research in set theory and the study of the infinite.