Glossary

5.2 The Axiom of Choice and its equivalents

3 min read•august 7, 2024

The is a powerful tool in set theory that allows us to make infinite selections. It states we can pick one element from each set in any collection of non-empty sets. This axiom is crucial but controversial due to its non-constructive nature.

The Axiom of Choice has several equivalent statements, including and the . These equivalences show the axiom's far-reaching implications in mathematics, from topology to algebra and beyond.

The Axiom of Choice and Equivalent Statements

Axiom of Choice and Its Formulation

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  • States that given any collection of non-empty sets, it is possible to select one element from each set to form a new set
  • Can be formalized as follows: For any set XX of non-empty sets, there exists a function f:X→⋃Xf:X\to\bigcup X such that for all A∈XA\in X, f(A)∈Af(A)\in A
  • Allows for the construction of sets by making an infinite number of arbitrary choices
  • Controversial axiom due to its non-constructive nature and counterintuitive consequences ()

Equivalent Statements to the Axiom of Choice

  • Zorn's Lemma states that if a partially ordered set PP has the property that every chain in PP has an upper bound in PP, then PP contains at least one maximal element
    • A chain is a totally ordered subset of PP
    • An upper bound of a subset SS of PP is an element b∈Pb\in P such that s≤bs\leq b for all s∈Ss\in S
  • Well-ordering Principle asserts that every set can be well-ordered
    • A well-ordering on a set SS is a total order on SS with the property that every non-empty subset of SS has a least element
  • for cardinals states that for any two κ\kappa and λ\lambda, exactly one of the following holds: κ<λ\kappa<\lambda, κ=λ\kappa=\lambda, or κ>λ\kappa>\lambda

Applications and Consequences of the Axiom of Choice

Tychonoff's Theorem and Its Implications

  • States that the product of any collection of compact topological spaces is compact
    • A topological space is compact if every open cover has a finite subcover
  • Relies on the Axiom of Choice for its proof in the infinite case
  • Has important applications in functional analysis (Alaoglu's theorem) and algebraic topology (compactness of profinite groups)

Hausdorff's Maximal Principle and Its Uses

  • States that every partially ordered set contains a maximal totally ordered subset (a chain)
  • Equivalent to the Axiom of Choice
  • Used in the proof of the existence of in rings and the existence of on sets

Axiomatic Set Theory

ZFC Set Theory and Its Axioms

  • Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) is a standard axiomatic system for set theory
  • Consists of the following axioms:
    1. : Two sets are equal if and only if they have the same elements
    2. : There exists a set with no elements, denoted by ∅\emptyset
    3. : If aa and bb are sets, then there exists a set {a,b}\{a,b\} containing exactly aa and bb
    4. : For any set XX, there exists a set ⋃X\bigcup X containing all elements of elements of XX
    5. : For any set XX, there exists a set P(X)\mathcal{P}(X) containing all subsets of XX
    6. : There exists an inductive set (a set containing ∅\emptyset and closed under the successor operation)
    7. : If FF is a function, then for any set XX, there exists a set {F(x):x∈X}\{F(x):x\in X\}
    8. : Every non-empty set has an ∈\in-minimal element
    9. Axiom of Choice: As stated above
  • ZFC provides a rigorous foundation for mathematics and avoids paradoxes like Russell's paradox

Key Terms to Review (26)

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 Dependent Choices: The Axiom of Dependent Choices states that for any non-empty set and any binary relation that is well-founded, there exists a sequence of choices that can be made from the set such that each choice depends on the previous one. This axiom is essential in establishing the ability to construct sequences in a way that echoes the principles of the Axiom of Choice but applies to situations where choices are made in a dependent manner.
Axiom of Empty Set: The axiom of empty set states that there exists a set that contains no elements, often denoted by the symbol ∅. This foundational concept is crucial in set theory as it establishes the existence of a set and serves as a building block for defining other sets and operations. The empty set plays a significant role in mathematical logic and has implications in various areas, including the Axiom of Choice and its equivalents.
Axiom of Extensionality: The axiom of extensionality states that two sets are considered equal if and only if they have the same elements. This principle is fundamental in set theory, ensuring that the identity of a set is determined solely by its members. It connects to various aspects of formal set theories, emphasizing the importance of object identity in mathematics.
Axiom of Infinity: The Axiom of Infinity is a foundational principle in set theory that asserts the existence of infinite sets. It posits that there is at least one set that contains the empty set and is closed under the operation of forming unions with singletons, ultimately leading to the construction of the natural numbers. This axiom is essential for the development of number theory and connects to other critical axioms and principles in mathematical logic.
Axiom of Pairing: The Axiom of Pairing states that for any two sets, there exists a set that contains exactly those two sets as its elements. This axiom is essential in set theory as it allows the construction of pairs, which can be further utilized in the formulation of more complex sets and structures. This axiom plays a vital role in the foundation of Zermelo-Fraenkel set theory and connects with concepts such as the Axiom of Choice, as it establishes a basis for the existence of pairs that can be selected from larger collections.
Axiom of Power Set: The Axiom of Power Set states that for any set, there exists a set of all its subsets, known as the power set. This axiom is foundational in set theory, as it allows for the construction of larger sets from existing sets and connects to the notions of cardinality and infinite sets, particularly when discussing the Zermelo-Fraenkel axioms and the Axiom of Choice.
Axiom of Regularity: The Axiom of Regularity, also known as the Axiom of Foundation, asserts that every non-empty set A contains an element that is disjoint from A. This principle ensures that sets cannot contain themselves and prevents the existence of certain paradoxical constructions within set theory. By establishing a foundational framework for sets, this axiom plays a crucial role in avoiding inconsistencies and supporting other axioms, like the Axiom of Choice and its equivalents.
Axiom of Union: The Axiom of Union states that for any set, there exists a set that contains exactly the elements of the elements of that set. This means that if you have a set whose members are themselves sets, you can create a new set that combines all those members into one single set. This axiom plays a crucial role in building the foundation of set theory, linking to the Zermelo-Fraenkel axioms and influencing the understanding of choices and functions in computer science.
Axiom Schema of Replacement: The Axiom Schema of Replacement is a principle in set theory that asserts if you have a set and a definable function, you can create a new set containing the images of the elements of the original set under that function. This axiom allows for the construction of new sets from existing ones, enhancing the power of set theory by ensuring that operations on sets can yield valid results. It plays a crucial role in formalizing mathematical constructs and connects deeply with the foundational Zermelo-Fraenkel axioms, as well as with concepts like the Axiom of Choice.
Banach-Tarski Paradox: The Banach-Tarski Paradox is a theorem in set theory and mathematical logic which states that it is possible to take a solid ball in three-dimensional space, divide it into a finite number of non-overlapping pieces, and then reassemble those pieces into two solid balls identical to the original. This paradox illustrates the counterintuitive consequences of the Axiom of Choice, which allows for the selection of elements from infinite sets, even when such selections lead to seemingly impossible outcomes.
Cardinal numbers: Cardinal numbers are a type of number used to denote the size or quantity of a set, indicating 'how many' elements are present. They provide a way to measure the size of sets in both finite and infinite contexts, making them essential in understanding different sizes of infinity and the structure of mathematical objects. The concept of cardinality is heavily influenced by foundational axioms and principles, which shape how we comprehend sets and their relationships.
Countable Axiom of Choice: The Countable Axiom of Choice states that for any countable collection of non-empty sets, it is possible to select exactly one element from each set. This principle is crucial in set theory as it allows for the construction of functions that map elements from a collection of sets to their respective choices, establishing a basis for many results in analysis and topology. The axiom is particularly significant when dealing with infinite sets, providing a way to handle selections that would otherwise be problematic.
Existence of a basis: The existence of a basis refers to the concept that for any vector space, there exists a set of vectors that are linearly independent and span the entire space. This notion is crucial in understanding the structure and dimensionality of vector spaces, as it guarantees that we can express any vector within the space as a unique linear combination of the basis vectors.
Filter bases: A filter base is a collection of sets that can be used to generate a filter on a given set, satisfying specific conditions. It is essential in topology and set theory, particularly when discussing the Axiom of Choice and its equivalents, as it provides a way to construct filters that can be applied in various mathematical contexts.
Giuseppe Peano: Giuseppe Peano was an Italian mathematician known for his work in mathematical logic and the foundations of mathematics, particularly for developing the Peano axioms, which define the natural numbers in terms of a set of axioms. His contributions laid the groundwork for formalizing mathematical theories and also intersect with the Axiom of Choice, as they both address foundational aspects of mathematics.
Hausdorff's Maximal Principle: Hausdorff's Maximal Principle states that in any partially ordered set, there exists a maximal element in every chain. A chain is a totally ordered subset, meaning that every pair of elements in the chain can be compared. This principle connects to the Axiom of Choice because it relies on choosing elements from potentially infinite sets to construct these chains and establish the existence of maximal elements.
Maximal ideals: Maximal ideals are specific types of ideals in a ring that are as large as possible without being equal to the whole ring. An ideal I in a ring R is maximal if it is proper (not equal to R) and there are no other ideals that contain I except for R itself. This concept plays a crucial role in understanding the structure of rings and relates to the Axiom of Choice, as well as Zorn's Lemma, which ensures the existence of maximal elements in partially ordered sets.
Non-measurable sets: Non-measurable sets are subsets of a given space that cannot be assigned a consistent measure using standard methods of measure theory. They often arise in contexts involving the Axiom of Choice, leading to paradoxical constructions that challenge our intuition about size and volume. Understanding these sets is crucial because they illustrate the limitations of measure theory and how certain assumptions can lead to contradictions.
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.
Trichotomy Principle: The Trichotomy Principle states that for any two elements, say a and b, one and only one of the following is true: either a is less than b, a is equal to b, or a is greater than b. This principle is foundational in the field of ordered sets and provides a clear framework for comparing elements within a set, ensuring that any two elements can be distinctly categorized in relation to one another.
Tychonoff's Theorem: Tychonoff's Theorem states that the product of any collection of compact topological spaces is compact in the product topology. This fundamental result in topology emphasizes the importance of compactness and the Axiom of Choice, as it relies on the ability to select open covers from each space involved in the product. The theorem highlights how compactness can be preserved through infinite products, which has significant implications for various areas in mathematics.
Ultrafilters: Ultrafilters are a special kind of filter on a set that have the property of being maximal, meaning they cannot be extended further without losing their filter characteristics. They help in the study of topology and set theory by providing a way to generalize the concept of convergence and limits. This makes ultrafilters significant when discussing the Axiom of Choice, as they relate to the selection of elements from sets and the ability to create certain types of mathematical structures.
Well-Ordering Principle: The well-ordering principle states that every non-empty set of natural numbers has a least element. This concept is crucial because it establishes a foundation for inductive reasoning and the structure of the natural numbers, connecting directly to total orders and the use of choice in mathematics.
ZFC Set Theory: ZFC set theory, short for Zermelo-Fraenkel set theory with the Axiom of Choice, is a foundational system for mathematics that defines the nature of sets and their relationships. It consists of a collection of axioms that formalize the properties and operations of sets, allowing mathematicians to rigorously discuss mathematical concepts. The inclusion of the Axiom of Choice is crucial as it asserts that for any set of non-empty sets, there exists a choice function that selects one element from each set.
Zorn's Lemma: Zorn's Lemma is a principle in set theory that states if every chain (totally ordered subset) in a partially ordered set has an upper bound, then the entire set contains at least one maximal element. This concept is crucial in many areas of mathematics and is closely related to the structure of partially ordered sets, the axioms of set theory, and various forms of the Axiom of Choice.
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