How To Break Absoluteness Of Power Set For Transitive Models?

In the realm of set theory and model theory, the concept of absoluteness plays a crucial role in understanding how formulas behave across different models. A formula is said to be absolute if its truth value remains consistent across various transitive models. However, certain fundamental operations in set theory, such as the power set operation, exhibit a fascinating characteristic: they are not absolute for transitive models. This article delves into the intricacies of this phenomenon, exploring why the formula representing the power set operation, denoted as $\phi (x,y):y=P(x)$\varphi(x, y) y = P(x), fails to be absolute. We will unpack the definitions, theorems, and examples that illuminate this cornerstone of set theory, making the topic accessible to both students and researchers. Understanding the limitations of absoluteness for the power set is not merely an academic exercise; it has profound implications for the consistency and independence results in set theory, particularly in the context of forcing and constructible sets.

Absoluteness in Model Theory

Before we dissect the power set absoluteness issue, it’s crucial to define what absoluteness means in the context of model theory. We say a formula ${\varphi(v_1, \dots, v_n)}$ in the language of set theory ${L_\in}$ is absolute for a class X of transitive models if, for any transitive models ${M}$ and ${N}$ in X, and for any elements ${a_1, \dots, a_n}$ in both ${M}$ and ${N}$, the formula ${\varphi}$ holds for these elements in ${M}$ if and only if it holds in ${N}$. Formally, this can be written as:

${[M \models \varphi(a_1, \dots, a_n)] \iff [N \models \varphi(a_1, \dots, a_n)]}$

Where ${M \models \varphi(a_1, \dots, a_n)}$ denotes that the formula ${\varphi}$ is true in the model ${M}$ when the free variables are interpreted as ${a_1, \dots, a_n}$. Transitive models are particularly important because they have the property that if an element ${x}$ is in the model ${M}$, then every element ${y}$ of ${x}$ is also in ${M}$. This characteristic simplifies many arguments and makes transitive models a natural setting for studying set-theoretic absoluteness.

The concept of absoluteness is essential in establishing consistency results and independence results in set theory. When a formula is absolute between a model and its submodels, it ensures that certain properties and relationships are preserved across different set-theoretic universes. This is particularly relevant when dealing with axioms and theorems that need to hold universally, irrespective of the specific model under consideration. For instance, the absoluteness of basic set operations, such as union, intersection, and pairing, guarantees that these operations behave consistently across transitive models. This consistency is vital for constructing and analyzing more complex set-theoretic structures.

However, not all formulas are absolute. The lack of absoluteness for certain formulas, such as the power set formula, reveals the subtle and sometimes counterintuitive nature of set theory. It highlights the fact that the universe of sets is not a monolithic entity but rather a complex hierarchy where certain constructions and relationships can vary depending on the model. This variability is not a flaw but a feature that allows for the rich tapestry of set-theoretic models and the possibility of exploring different set-theoretic axioms and their consequences. By understanding which formulas are absolute and which are not, we gain deeper insights into the foundations of mathematics and the structure of the set-theoretic universe.

Examples of Absolute Formulas

Many basic set-theoretic formulas are absolute for transitive models. For example, formulas expressing membership (${x \in y}$), equality (${x = y}$), the empty set (${x = \emptyset}$), pairing (${z = \{x, y\}}$), union (${z = x \cup y}$), and intersection (${z = x \cap y}$) are all absolute. These formulas involve fundamental operations and relations that are essential for constructing and manipulating sets. Their absoluteness ensures that these basic building blocks of set theory behave consistently across different models.

To illustrate, let's consider the formula for the empty set, ${\varphi(x) : x = \emptyset}$. This formula states that ${x}$ is the empty set, which means that there is no ${y}$ such that ${y \in x}$. Formally, this can be expressed as:

${\varphi(x) : \forall y (y \notin x)}$

Now, suppose we have two transitive models, ${M}$ and ${N}$, and an element ${a}$ that is in both ${M}$ and ${N}$. If ${M \models \varphi(a)}$, then ${a}$ is the empty set in ${M}$, meaning there is no element in ${M}$ that is a member of ${a}$. Since ${N}$ is also a transitive model, if ${a}$ is the empty set in ${M}$, it must also be the empty set in ${N}$. Therefore, ${N \models \varphi(a)}$. Conversely, if ${N \models \varphi(a)}$, then ${a}$ is the empty set in ${N}$, and similarly, it must be the empty set in ${M}$. Hence, ${M \models \varphi(a)}$. This shows that the formula for the empty set is absolute for transitive models.

Similar arguments can be made for the other basic set-theoretic formulas mentioned above. The absoluteness of these formulas is crucial because they form the foundation upon which more complex set-theoretic concepts and constructions are built. Without this absoluteness, it would be challenging to reason consistently about sets and their properties across different models.

The Power Set Operation and Its Non-Absoluteness

The power set operation, denoted by ${P(x)}$, is a fundamental concept in set theory. The power set of a set ${x}$ is the set of all subsets of ${x}$. Formally,

${P(x) = \{y : y \subseteq x\}}$

The formula ${\varphi(x, y) : y = P(x)}$ states that ${y}$ is the power set of ${x}$. This formula is surprisingly not absolute for transitive models. This non-absoluteness arises from the fact that the notion of “subset” can vary between different transitive models. While the elements of ${x}$ that exist within a transitive model ${M}$ are well-defined, the subsets of ${x}$ that are included in ${M}$ may not comprise all the subsets of ${x}$. In other words, a transitive model ${M}$ might not be “aware” of all the subsets of ${x}$. This subtle distinction leads to the failure of absoluteness for the power set formula.

To understand this better, consider two transitive models, ${M}$ and ${N}$, with ${M \subseteq N}$. Suppose ${x}$ is an element in both ${M}$ and ${N}$. Let ${y_M}$ be the power set of ${x}$ in ${M}$, and let ${y_N}$ be the power set of ${x}$ in ${N}$. That is,

${y_M = \{z \in M : z \subseteq x\}\text{ in } M}$

${y_N = \{z \in N : z \subseteq x\}\text{ in } N}$

Even if ${y_M}$ and ${y_N}$ are both sets, it is possible that ${y_M \neq y_N}$. Specifically, ${y_M}$ is the set of all subsets of ${x}$ that are also elements of ${M}$, while ${y_N}$ is the set of all subsets of ${x}$ that are elements of ${N}$. Since ${N}$ might contain subsets of ${x}$ that are not in ${M}$, it is possible that ${y_N}$ contains elements that are not in ${y_M}$.

This discrepancy illustrates that the power set operation is relative to the model in which it is computed. The formula ${\varphi(x, y) : y = P(x)}$ is true in a model ${M}$ if ${y}$ contains all subsets of ${x}$ that are also elements of ${M}$. However, this does not guarantee that ${y}$ contains all subsets of ${x}$ in the broader universe of sets, especially when considering a larger model ${N}$. This relativity is what causes the non-absoluteness of the power set formula.

Example Illustrating Non-Absoluteness

To make this concept more concrete, let's consider a specific example. Suppose we have a transitive model ${M}$ that does not satisfy the full power set axiom. This means there exists a set ${x}$ in ${M}$ such that the actual power set of ${x}$, denoted by ${\mathcal{P}(x)}$, is not an element of ${M}$. In other words, while ${M}$ contains some subsets of ${x}$, it does not contain all of them.

Now, let's consider a larger transitive model ${N}$ that extends ${M}$ and does satisfy the power set axiom. This means that for any set ${x}$ in ${N}$, the power set ${\mathcal{P}(x)}$ is also an element of ${N}$. Suppose ${x}$ is the same set as before, which is in both ${M}$ and ${N}$.

In ${M}$, the power set of ${x}$, as computed within ${M}$, is ${P_M(x) = \{y \in M : y \subseteq x\}}$. This set contains only those subsets of ${x}$ that are also elements of ${M}$. However, since ${M}$ does not contain the full power set of ${x}$, ${P_M(x)}$ is not the same as ${\mathcal{P}(x)}$.

In ${N}$, the power set of ${x}$, as computed within ${N}$, is ${P_N(x) = \{y \in N : y \subseteq x\}}$. Since ${N}$ satisfies the power set axiom, ${P_N(x)}$ is the full power set of ${x}$, i.e., ${P_N(x) = \mathcal{P}(x)}$.

Thus, in ${M}$, the formula ${\varphi(x, P_M(x))}$ is true, as ${P_M(x)}$ is considered the power set of ${x}$ within ${M}$. However, in ${N}$, the formula ${\varphi(x, P_M(x))}$ is false because ${P_M(x)}$ is not the actual power set of ${x}$ in the broader universe of sets represented by ${N}$. Instead, the true power set of ${x}$ in ${N}$ is ${P_N(x)}$, and ${P_N(x) \neq P_M(x)}$.

This example clearly demonstrates the non-absoluteness of the power set formula. The same set ${x}$ has different “power sets” depending on the model in which the operation is performed. This discrepancy is a direct consequence of the fact that models can have different notions of what constitutes a “subset,” especially when dealing with infinite sets. The failure of the power set formula to be absolute is a key insight in set theory and has significant implications for various advanced topics.

Implications of Non-Absoluteness

The non-absoluteness of the power set operation has profound implications for various aspects of set theory, especially concerning the consistency and independence of axioms. One of the most significant areas where this non-absoluteness plays a role is in the context of the Generalized Continuum Hypothesis (GCH) and the Continuum Hypothesis (CH).

The Continuum Hypothesis, first proposed by Georg Cantor, states that there is no set whose cardinality is strictly between that of the integers and that of the real numbers. In other words, it asserts that there is no set with cardinality strictly between ${\aleph_0}$ (the cardinality of the natural numbers) and ${2^{\aleph_0}}$ (the cardinality of the power set of the natural numbers, which is equal to the cardinality of the real numbers). The Generalized Continuum Hypothesis extends this idea to all cardinals, stating that for any infinite cardinal ${\kappa}$, there is no cardinal strictly between ${\kappa}$ and ${2^\kappa}$.

The non-absoluteness of the power set operation is crucial in understanding why the CH and GCH are independent of the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). Independence means that neither CH nor its negation can be proven from the axioms of ZFC. This independence was famously demonstrated by Kurt Gödel and Paul Cohen.

Gödel showed that if ZFC is consistent, then ZFC + CH is also consistent. He constructed a model of set theory, known as the constructible universe ${L}$, in which the CH (and the GCH) holds. The constructible universe is a minimal model of set theory, built up iteratively from the empty set using only definable sets. The absoluteness properties within ${L}$ play a critical role in Gödel’s proof. However, even within ${L}$, the non-absoluteness of the power set operation requires careful handling to ensure that the construction remains consistent.

Cohen, on the other hand, proved that if ZFC is consistent, then ZFC + ¬CH is also consistent. His method, known as forcing, involves constructing a new model of set theory by adding sets to an existing model. The forcing technique crucially relies on the non-absoluteness of the power set operation. By carefully adding sets to a model, Cohen was able to manipulate the cardinality of the continuum (${2^{\aleph_0}}$) without changing the cardinality of ${\aleph_0}$, thus violating the CH. The non-absoluteness of the power set ensures that the forcing extensions can change the power set of a set without altering the set itself, which is essential for Cohen’s proof.

The significance of the non-absoluteness of the power set also extends to other independence results in set theory. Many statements in set theory are independent of ZFC, and the techniques used to prove these independence results often rely on the ability to manipulate power sets in a controlled manner. This manipulation is only possible because the power set operation is not absolute, allowing us to construct models with different power sets and, consequently, different set-theoretic properties.

In summary, the non-absoluteness of the power set operation is not merely a technical curiosity; it is a fundamental feature of set theory that underpins our understanding of the limitations of the axiomatic method and the diversity of set-theoretic universes. It allows for the exploration of different set-theoretic possibilities and is essential for the development of advanced techniques in set theory, such as forcing and the construction of constructible sets.

Connections to Forcing and Constructible Sets

The non-absoluteness of the power set is intricately linked to the techniques of forcing and the concept of constructible sets, both of which are crucial in establishing independence results in set theory. Forcing, developed by Paul Cohen, is a method for extending a model of set theory to create a new model with specific desired properties. Constructible sets, formalized by Kurt Gödel, form a minimal inner model of set theory in which the Axiom of Choice and the Continuum Hypothesis hold.

In the context of forcing, the non-absoluteness of the power set operation is fundamental to how forcing extensions alter the set-theoretic universe. The basic idea behind forcing is to start with a ground model ${M}$ and add a new set ${G}$, called a generic filter, to obtain a larger model ${M[G]}$. The sets in the forcing extension ${M[G]}$ are constructed using names in the ground model ${M}$, and the properties of the generic filter ${G}$ are carefully controlled to ensure that the extended model satisfies the axioms of ZFC.

The critical aspect here is that the power set of a set in the ground model ${M}$ can be different in the forcing extension ${M[G]}$. This difference is a direct consequence of the non-absoluteness of the power set operation. When a generic filter ${G}$ is added, it introduces new subsets that were not present in the ground model ${M}$. As a result, the power set of a set ${x}$ in ${M[G]}$ may contain subsets that are not in the power set of ${x}$ in ${M}$. This manipulation of power sets is central to the forcing technique, as it allows us to control the cardinalities and other properties of the sets in the extended model.

For example, Cohen’s proof of the independence of the Continuum Hypothesis relies on forcing to add a set of real numbers without adding new countable sets. This addition changes the cardinality of the continuum in the extended model, demonstrating that the CH can be false in some models of ZFC. The non-absoluteness of the power set ensures that this manipulation can be done consistently, without collapsing cardinals or violating the axioms of ZFC.

On the other hand, the constructible universe ${L}$ is a class of sets built up in a well-defined manner, where at each stage, we add only the sets that are definable from the sets constructed in the previous stages. The constructible universe is a transitive inner model of ZFC, meaning it is a model of ZFC that contains all the ordinal numbers. Gödel showed that the Axiom of Choice and the Generalized Continuum Hypothesis hold in ${L}$.

The absoluteness properties of formulas within ${L}$ are crucial for proving these results. In particular, the constructible power set operation, which computes the power set of a set within ${L}$, is absolute with respect to ${L}$. This means that if ${x}$ is a set in ${L}$, then the power set of ${x}$ as computed within ${L}$ is the same regardless of which larger model of set theory we consider. However, even within ${L}$, the non-absoluteness of the full power set operation (i.e., the power set operation as computed in the entire universe of sets) requires careful consideration. The constructible universe is constructed in such a way that it contains only those sets that are necessary to satisfy the axioms of ZFC and the definability conditions, which helps in maintaining the consistency of the model.

In summary, the non-absoluteness of the power set operation is a critical link between forcing and the constructible universe. Forcing exploits this non-absoluteness to create models where the Continuum Hypothesis fails, while the construction of the constructible universe relies on controlled absoluteness properties to create a model where the Continuum Hypothesis holds. These techniques, rooted in the non-absoluteness of the power set, provide powerful tools for exploring the landscape of set-theoretic universes and establishing the independence of fundamental set-theoretic statements.

Conclusion

The non-absoluteness of the power set operation for transitive models is a cornerstone concept in advanced set theory and model theory. This phenomenon arises from the fact that the power set of a set can differ across models, depending on which subsets are included within the model's universe. The formula ${\varphi(x, y) : y = P(x)}$, representing the power set operation, is not absolute because a model may not contain all the subsets of a given set, leading to discrepancies in how the power set is computed across different models. This non-absoluteness has profound implications for the independence results in set theory, particularly concerning the Continuum Hypothesis and the Generalized Continuum Hypothesis.

The independence of the Continuum Hypothesis from the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) is a landmark result, with Kurt Gödel demonstrating the consistency of ZFC + CH and Paul Cohen establishing the consistency of ZFC + ¬CH. Both Gödel’s constructible universe ${L}$ and Cohen’s forcing technique rely heavily on the properties of absoluteness and non-absoluteness, respectively. The non-absoluteness of the power set operation is crucial for forcing, as it allows for the manipulation of cardinalities by adding sets to a model without altering the original sets. This manipulation is essential for constructing models where the Continuum Hypothesis fails. In contrast, the constructible universe ${L}$ leverages the absoluteness of constructible power sets to ensure that the CH holds within this minimal model.

Furthermore, the non-absoluteness of the power set operation extends its influence to other independence results in set theory. Many statements in set theory are independent of ZFC, and the techniques used to prove these results often depend on the controlled manipulation of power sets. This manipulation is only possible because the power set operation is not absolute, enabling the construction of models with diverse set-theoretic properties.

In conclusion, understanding the non-absoluteness of the power set operation is vital for comprehending the landscape of set-theoretic universes and the limitations of the axiomatic method. It is not merely a technical detail but a fundamental feature that underpins the power and richness of modern set theory. By delving into these intricacies, we gain deeper insights into the foundations of mathematics and the nature of the infinite, appreciating the subtle yet profound role of absoluteness in shaping our understanding of sets and models. The non-absoluteness of the power set, therefore, remains a central theme in set-theoretic research, continuing to inspire new discoveries and deepen our appreciation of the complexity and beauty of set theory.