10.3 Forcing and the independence of CH
2 min read•august 7, 2024
is a game-changing technique in set theory. It lets us build new models of math, proving some ideas can't be settled using standard rules. This is huge for understanding what we can and can't prove.
used forcing to show the Continuum Hypothesis is independent. This means we can't prove or disprove it using normal set theory. It's a mind-bending result that shook up math.
Forcing and Independence
Paul Cohen's Contributions to Set Theory
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- Paul Cohen introduced the technique of forcing in set theory
- Proved the of the Continuum Hypothesis (CH) from with the (ZFC)
- Showed that both CH and its negation are consistent with ZFC
- Received the Fields Medal in 1966 for his groundbreaking work in set theory
The Forcing Technique
- Forcing is a method for extending models of set theory
- Adds new sets to a model while preserving the truth of certain statements
- Allows for the construction of models with desired properties
- Used to prove independence results and explore the consistency of various statements in set theory
Independence and Undecidability
- Independence refers to a statement being neither provable nor disprovable within a given axiom system
- The Continuum Hypothesis is independent of ZFC
- Independence proofs demonstrate the limitations of an axiom system
- means that a statement cannot be proven or disproven within a given formal system (Gödel's Incompleteness Theorems)
Models and Extensions
Generic Extensions and Models
- are models of set theory obtained by applying the forcing technique
- Begin with a (M) and a (P) in M
- Construct a (G) on P, which is a subset of P meeting certain conditions
- The (M[G]) is the smallest model containing M and G
- M[G] satisfies ZFC if M does
The Axiom of Constructibility and Its Negation
- The , denoted as V=L, states that every set is constructible
- are those that can be built from simpler sets using certain operations
- V represents the universe of all sets, and L represents the
- The (V≠L) asserts that there are sets that are not constructible
- Cohen's forcing technique can be used to construct models where V≠L holds
Advanced Set Theory
Large Cardinals and Their Implications
- are cardinal numbers with certain strong properties
- Examples include , , and
- The existence of large cardinals cannot be proven in ZFC alone
- Large cardinal axioms extend ZFC and have significant consequences for the structure of the set-theoretic universe
- The study of large cardinals has led to a rich hierarchy of consistency strength and implications between various set-theoretic statements
Key Terms to Review (32)
Axiom of Choice: The Axiom of Choice states that for any collection of non-empty sets, there exists a way to select one element from each set, even if there is no explicit rule for making the selection. This concept is fundamental in set theory and connects various results and theorems across different areas of mathematics.
Axiom of Constructibility: The Axiom of Constructibility (V = L) states that every set is constructible, meaning that every set can be built up in a systematic way from simpler sets. This axiom has significant implications for the foundations of set theory and directly relates to the independence of the Continuum Hypothesis and Gödel's constructible universe, which show how certain mathematical truths can depend on the acceptance of this axiom.
Back-and-forth argument: A back-and-forth argument is a method used in mathematical logic and set theory to establish the independence of certain statements, showing that both a statement and its negation can be consistently added to a given model. This technique is particularly useful in the context of proving that specific propositions, like the Continuum Hypothesis, cannot be proven or disproven using standard axioms. It highlights the idea that within set theory, some truths transcend the capabilities of formal proofs and axiomatic systems.
Cardinal arithmetic: Cardinal arithmetic is the branch of mathematics that deals with the addition, subtraction, multiplication, and division of cardinal numbers, which represent the size or quantity of sets. It plays a crucial role in understanding the nature of infinity and the relationships between different infinite sets, especially in the context of set theory and various axioms like the Continuum Hypothesis.
Cardinality: Cardinality refers to the measure of the 'number of elements' in a set, providing a way to compare the sizes of different sets. This concept allows us to classify sets as finite, countably infinite, or uncountably infinite, which is essential for understanding the structure of mathematical systems and their properties.
Constructible Sets: Constructible sets are collections of sets that can be explicitly defined and built up using a process that involves the cumulative hierarchy of sets. This idea is central to understanding Gödel's constructible universe, where every set can be constructed in a systematic way, leading to the exploration of consistency and independence results, particularly regarding the Continuum Hypothesis (CH). Constructible sets serve as a framework to analyze what can be known and established within set theory.
Constructible Universe: The constructible universe, often denoted as $L$, is a class of sets that can be constructed in a specific manner through a hierarchy of stages, using definable operations and previously constructed sets. This concept is vital in set theory as it helps to understand the consistency and independence of various axioms, such as the Axiom of Choice and the Continuum Hypothesis.
Countable: A set is called countable if its elements can be put into a one-to-one correspondence with the natural numbers, meaning that there exists a way to list all the elements of the set without missing any. Countable sets can be either finite or infinite, and understanding this concept is essential for discussing the properties of cardinal numbers and the implications of various axioms in set theory, especially in terms of determining the size of different types of infinite sets.
Elementary embedding: An elementary embedding is a type of function between two structures in model theory that preserves the truth of first-order statements. This means if a statement is true in one structure, it remains true in the other when the embedding is applied. Elementary embeddings are crucial for understanding relationships between different models and play a significant role in the study of forcing and the independence of various mathematical propositions.
Forcing: Forcing is a technique used in set theory to extend a given model of set theory to create a new model where certain statements hold true, particularly in proving the independence of various mathematical propositions. This method allows mathematicians to show that certain axioms, like the Continuum Hypothesis, can be independent of Zermelo-Fraenkel set theory, thus demonstrating their consistency or inconsistency.
Generic extension: A generic extension is a new model of set theory obtained by adding new sets in a controlled way through a process known as forcing. This concept plays a crucial role in demonstrating the independence of certain mathematical statements, like the Continuum Hypothesis, by showing that the properties of the original model can be preserved or altered based on the new sets added. In essence, a generic extension allows mathematicians to construct models where specific properties hold true while still maintaining consistency with established axioms.
Generic extensions: Generic extensions are models of set theory that result from a specific method called forcing, which is used to create new models by adding 'generic' sets to an existing model. This process allows mathematicians to explore various properties of models, such as the independence of certain statements like the Continuum Hypothesis (CH), as it creates models where specific conditions hold or fail. Understanding generic extensions is crucial for grasping how forcing alters the landscape of set theory.
Generic filter: A generic filter is a mathematical construct used in set theory and forcing to create models of set theory by systematically extending a given model with new sets. It acts as a way to add elements while preserving certain properties, enabling the exploration of different mathematical universes and their relationships, particularly in proving the independence of various set-theoretical statements such as the Continuum Hypothesis.
Ground model: A ground model is a set-theoretical universe that serves as the starting point for discussions about mathematical structures and their properties. It provides the foundational context in which the axioms of set theory are applied, particularly when analyzing concepts like forcing and the independence of certain propositions, such as the Continuum Hypothesis (CH). Understanding ground models is crucial for grasping how extensions of these models can affect the truth values of mathematical statements.
Inaccessible cardinals: Inaccessible cardinals are a type of large cardinal that cannot be reached by certain set-theoretic operations, such as taking power sets or forming unions of fewer than their own size. They are significant because they serve as a boundary for the consistency of various mathematical statements, including the Continuum Hypothesis. These cardinals are often used in advanced set theory to investigate the nature of infinity and the foundations of mathematics.
Independence: Independence refers to a property of mathematical statements or axioms where a statement cannot be proven or disproven using a given set of axioms. This concept is essential in understanding the limits of formal systems, as it illustrates that certain truths exist beyond provability within those systems. Recognizing independence helps clarify relationships between axioms and the statements they encompass, which is pivotal in exploring consistency and the foundations of set theory.
Independence of CH: The independence of the Continuum Hypothesis (CH) refers to the mathematical assertion that there is no set whose cardinality is strictly between that of the integers and the real numbers. This concept is significant because it was shown that both the CH and its negation can be consistent with the standard axioms of set theory if Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) is assumed, meaning that it cannot be proven or disproven using these axioms alone.
Kurt Gödel: Kurt Gödel was a renowned mathematician and logician, best known for his incompleteness theorems which revealed limitations in formal mathematical systems. His work established critical insights into the consistency and independence of axioms, influencing foundational aspects of mathematics and set theory.
Large cardinals: Large cardinals are certain types of infinite cardinal numbers that possess strong and significant properties, often extending the standard hierarchy of set theory. They are crucial in understanding the foundations of mathematics, especially in relation to consistency and independence results, such as those surrounding the Continuum Hypothesis. These cardinals imply the existence of large sets that cannot be constructed or defined within standard set theory frameworks.
Measurable cardinals: Measurable cardinals are a special type of large cardinal, which is a cardinal number that is so large that it cannot be reached by any standard set-theoretic operations. They have the property of being 'measurable', meaning there exists a non-trivial elementary embedding from the cardinal into itself, which preserves the structure of sets. This property connects them deeply with concepts like the continuum hypothesis and the consistency of various set-theoretic statements, making them crucial in contemporary research within the field.
Model construction: Model construction is a method used in set theory, particularly in the context of forcing, to create specific mathematical models that satisfy certain properties or axioms. This process involves defining a structure that can demonstrate the independence of particular propositions from standard set-theoretic axioms, such as the Continuum Hypothesis (CH). By constructing these models, mathematicians can explore the relationships between different mathematical statements and their truth values in varying contexts.
Negation of the Axiom of Constructibility: The negation of the axiom of constructibility states that there exist sets that cannot be constructed from simpler sets using the standard set-theoretic operations. This concept is significant in understanding the nature of mathematical sets and their relationships to cardinality, particularly in relation to the Continuum Hypothesis, where it suggests that there are larger infinities beyond those constructed under the axiom of constructibility.
Partial order: A partial order is a binary relation over a set that is reflexive, antisymmetric, and transitive. This means that in a partial order, some elements can be compared to one another while others cannot, creating a hierarchy or structure among them. The concept of partial orders is essential in various areas, including mathematics and computer science, as it helps to organize objects in a way that captures their relationships and properties.
Partially ordered set: A partially ordered set, or poset, is a set combined with a relation that allows for the comparison of some pairs of elements but not necessarily all. In a poset, the relation is reflexive, antisymmetric, and transitive, meaning each element can be compared to itself, if one element is less than another and vice versa, then they are equal, and if one element is less than a second and that second is less than a third, then the first is less than the third. This concept is crucial for understanding more complex structures in mathematics and plays a key role in various proofs and theories.
Paul Cohen: Paul Cohen was a prominent American mathematician known for his groundbreaking work in set theory and logic, particularly in demonstrating the independence of the Continuum Hypothesis from Zermelo-Fraenkel set theory with the Axiom of Choice. His innovative method of forcing transformed how mathematicians approached the foundations of set theory and significantly influenced subsequent research directions in the field.
Supercompact cardinals: Supercompact cardinals are a special type of large cardinal in set theory, which possess strong combinatorial properties. A cardinal is considered supercompact if, for every set of smaller cardinality, there is an elementary embedding into a larger structure that reflects certain properties of the original set. This concept is deeply connected to various areas in set theory, particularly in discussing the independence of the Continuum Hypothesis and exploring contemporary research directions.
Transitive model: A transitive model is a specific type of mathematical structure used in set theory where every element of a set is also a subset of that set, ensuring that the model exhibits transitive properties. This characteristic is crucial when discussing the foundations of mathematics, particularly in the context of constructions like forcing, which help establish the independence of various propositions from standard axioms.
Ultrapower: Ultrapower is a construction in set theory that allows for the creation of a new structure by taking a Cartesian product of a model of set theory and collapsing it using an ultrafilter. This process can reveal properties of models that are not immediately visible, especially in the context of non-standard analysis and forcing, leading to insights about the independence of certain mathematical statements.
Undecidability: Undecidability refers to the property of a decision problem where no algorithm can be constructed that will always lead to a correct yes-or-no answer for every possible input. This concept is crucial in mathematical logic and theoretical computer science, as it helps us understand the limitations of formal systems and the nature of certain propositions, especially concerning the Continuum Hypothesis and its independence from standard set theory.
Universe of Sets: The universe of sets is the collection that contains all possible sets under consideration in a particular context. It serves as the foundational backdrop for set theory, establishing boundaries within which operations and relations among sets are explored. This concept is crucial in understanding how different sets interact and how properties can be defined based on their relationships to the universe.
Zermelo-Fraenkel Set Theory: Zermelo-Fraenkel Set Theory (ZF) is a foundational system for mathematics that uses sets as the basic building blocks, formalized by a collection of axioms that dictate how sets behave and interact. This theory serves as a framework for discussing concepts such as infinity, ordinals, and the continuum hypothesis, while also addressing paradoxes in set theory and providing a rigorous basis for mathematical reasoning.
ZFC Consistency: ZFC consistency refers to the idea that the axioms of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) do not lead to any contradictions. This is important because if ZFC is consistent, it means that all theorems derived from it can be accepted as true within that framework. The consistency of ZFC is crucial for discussing properties such as the Continuum Hypothesis (CH) and understanding what can be proven or disproven within this foundational system.