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Axiom of empty set
Axiom of empty set
Axiom of empty set
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In axiomatic set theory, the axiom of empty set,[1][2] also called the axiom of null set[3] and the axiom of existence,[4][5] is a statement that asserts the existence of a set with no elements.[3] It is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrable truth in Zermelo set theory and Zermelo–Fraenkel set theory, with or without the axiom of choice.[6]

Formal statement

[edit]

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

.[1][2][5]

Or, alternatively, .[7]

In words:

There is a set such that no element is a member of it.

Interpretation

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We can use the axiom of extensionality to show that there is only one empty set. Since it is unique we can name it. It is called the empty set (denoted by { } or ∅). The axiom, stated in natural language, is in essence:

An empty set exists.

This formula is a theorem and considered true in every version of set theory[8]. The only controversy is over how it should be justified: by making it an axiom; by deriving it from a set-existence axiom (or logic) and the axiom of separation; by deriving it from the axiom of infinity; or some other method.

In some formulations of ZF, the axiom of empty set is actually repeated in the axiom of infinity. However, there are other formulations of that axiom that do not presuppose the existence of an empty set. The ZF axioms can also be written using a constant symbol representing the empty set; then the axiom of infinity uses this symbol without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty.

Furthermore, one sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set may still be required. However, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the axiom schema of separation. This is true, since the empty set is a subset of any set consisting of those elements that satisfy a contradictory formula.

In many formulations of first-order predicate logic, the existence of at least one object is always guaranteed. If the axiomatization of set theory is formulated in such a logical system with the axiom schema of separation as axioms, and if the theory makes no distinction between sets and other kinds of objects (which holds for ZF, KP, and similar theories), then the existence of the empty set is a theorem.

If separation is not postulated as an axiom schema but derived as a theorem schema from the schema of replacement (as is sometimes done), the situation is more complicated and depends on the exact formulation of the replacement schema. The formulation used in the axiom schema of replacement article only allows to construct the image F[a] when a is contained in the domain of the class function F; then the derivation of separation requires the axiom of empty set. On the other hand, the constraint of totality of F is often dropped from the replacement schema, in which case it implies the separation schema without using the axiom of empty set (or any other axiom for that matter).

References

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Further reading

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Axiom of empty set

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The Axiom of the Empty Set, also known as the Axiom, is a core axiom in Zermelo-Fraenkel set theory (ZF) that asserts the existence of a unique set with no elements, providing the foundational building block for all sets in the theory. Formally stated as ∃x∀y(y∉x)\exists x \forall y (y \notin x), it guarantees a set ∅\varnothing such that no object belongs to it, and its uniqueness follows from the , which equates sets with identical members. Introduced by in his 1908 axiomatization of as part of the Axiom of Elementary Sets (Axiom II), it addressed paradoxes in by restricting set formation to well-defined operations starting from the . This axiom enables the construction of all natural numbers and other sets via operations like and union, forming the basis of the von Neumann cumulative hierarchy where sets are built iteratively from ∅\varnothing. Although derivable in full ZF from the and the of Separation—by taking the intersection of an with a contradictory property—it is explicitly included in many formulations for clarity, especially in systems lacking or for pedagogical purposes.

Formal Definition

Statement in Zermelo-Fraenkel Set Theory

In Zermelo-Fraenkel set theory (ZF), the axiom of the asserts the existence of a set that contains no elements. Formally, it is stated in as ∃x ∀y (y∉x),\exists x \, \forall y \, (y \notin x), where ∃x\exists x is the existential quantifier indicating that there exists at least one such set xx, ∀y\forall y is the universal quantifier ranging over all sets yy, and ∉\notin denotes the negation of set membership ∈\in. This formulation ensures that the universe of sets includes at least one set devoid of members, foundational for constructing further sets via other axioms. The notation ∅\varnothing is conventionally used to denote this empty set, symbolizing its unique role as the set with no elements. The quantifiers in the axiom operate within the language of , where predicates like ∈\in define membership relations, and the axiom's assertion prevents a purely "pure" set universe without basic building blocks. From this , combined with the , it follows that the is unique: there exists exactly one such set in the ZF universe.

Equivalent Formulations

One equivalent formulation of the axiom of the empty set in Zermelo-Fraenkel set theory emphasizes its uniqueness, derived from the axiom of extensionality: there exists a unique set xx such that no set yy satisfies y∈xy \in x. This captures the empty set as the sole set containing neither itself nor any other set as an element. A related characterization, leveraging concepts from the axiom of pairing (which allows construction of sets like singletons {a}={a,a}\{a\} = \{a, a\}), expresses the empty set indirectly through the foundational role it plays in building such structures without assuming elements. Specifically, the empty set serves as the base case for inductive constructions, ensuring the existence of sets with precisely zero elements, distinct from paired or singleton sets that contain at least one. In type-theoretic systems, such as , an analogous formulation introduces the as a type with no introduction rules or constructors, meaning it admits no terms or inhabitants. This empty type functions equivalently to the by representing a "set" devoid of members, providing a constructive foundation where the absence of elements is primitive rather than asserted via existence. To demonstrate equivalence to the standard ZF formulation ∃x ∀y (y∉x)\exists x \, \forall y \, (y \notin x), consider a proof sketch in , where predicates can quantify over . Assume the existence of a set with no members, formalized as ∃x ∀P (P⊆x→P=∅)\exists x \, \forall P \, (P \subseteq x \to P = \emptyset), where PP ranges over ; this implies xx has no elements, as any non-empty would contradict the universal emptiness. Conversely, from the ZF statement, ensures uniqueness, and second-order quantification over the empty membership relation ∈\in yields the same vacuous condition, confirming .

Role in Set Theory Foundations

Justification for Including the Axiom

The axiom of the empty set is essential in (ZF) because, without it and without the , the other axioms—such as , , union, , replacement, and foundation—presuppose the existence of at least one set to operate but fail to assert any initial set, resulting in a theory where no sets can be constructed or proven to exist. This would render the theory incapable of modeling any mathematical structures. However, in full ZF including the and the axiom schema of separation, the can be derived: the asserts the existence of an SS, and separation yields the {x∈S∣x≠x}\{x \in S \mid x \neq x\}, which contains no elements and is thus empty. Despite this derivability, the is explicitly included in many formulations for clarity, especially in systems lacking infinity (such as ) or for pedagogical purposes to emphasize the starting point of set constructions. A primary justification for including the axiom is its role as the foundational starting point for the cumulative of sets, known as the VV. The hierarchy begins with V0=∅V_0 = \emptyset, the , which serves as the base level from which all subsequent levels are iteratively constructed: Vα+1=P(Vα)V_{\alpha+1} = \mathcal{P}(V_\alpha) for successor stages and unions at limit ordinals. This ensures that every set in the universe appears at some rank, providing a well-ordered structure for set-theoretic constructions and aligning with the iterative conception of sets. Without ∅\varnothing, the hierarchy cannot initiate in formulations without derivation from infinity. The axiom further enables the explicit construction of finite sets through interactions with other basic axioms, such as power set and union. For instance, the power set of ∅\emptyset yields {∅}\{\emptyset\}, the singleton containing the empty set, which can then be used with the pairing axiom to form {∅,{∅}}\{\emptyset, \{\emptyset\}\} and subsequently all finite von Neumann ordinals representing natural numbers. This step-by-step buildup is crucial for foundational mathematics, as it guarantees the existence of finite structures independently of the axiom of infinity.

Interactions with Other Axioms

The uniqueness of the empty set relies fundamentally on the , which states that two sets are equal if and only if they have precisely the same elements: ∀x∀y(∀z(z∈x↔z∈y)→x=y)\forall x \forall y \bigl( \forall z (z \in x \leftrightarrow z \in y) \to x = y \bigr). To see this, suppose aa and bb are sets with no elements, so ∀z ¬(z∈a)\forall z \, \neg (z \in a) and ∀z ¬(z∈b)\forall z \, \neg (z \in b). Then, for all zz, z∈a↔z∈bz \in a \leftrightarrow z \in b holds vacuously, since both sides are false. By , a=ba = b, ensuring there is at most one . Without extensionality, multiple distinct "empty" collections could exist, undermining the foundational equality of sets determined solely by membership. (Jech, Set Theory, 2003) The interacts closely with the , which asserts that for any set xx, there exists a set yy such that every element of yy is an element of some member of xx: ∀x∃y∀z(z∈y↔∃w(z∈w∧w∈x))\forall x \exists y \forall z \bigl( z \in y \leftrightarrow \exists w (z \in w \land w \in x) \bigr). When x=∅x = \emptyset, the right-hand side ∃w(z∈w∧w∈∅)\exists w (z \in w \land w \in \emptyset) is false for all zz, since no w∈∅w \in \emptyset. Thus, y=∅y = \emptyset satisfies the condition, demonstrating that the union of the is itself empty. This synergy ensures that the behaves consistently under union operations, allowing derivations of empty unions in contexts without contributing elements, such as initial stages in the cumulative . (Jech, , 2003) Although derivable in full ZF from the axioms of and separation, the empty set axiom is independent of the remaining ZF excluding those two. For example, models of the of , , union, , replacement, and foundation can be constructed without an or any sets at all (the empty model satisfies them vacuously). These results confirm that the empty set axiom cannot be derived from the non-, non-separation ZF .

Historical Development

Origins in Zermelo's Axiomatization

In the early 20th century, the foundations of set theory were shaken by paradoxes such as Bertrand Russell's paradox, discovered in 1901 and published in 1903, which revealed inconsistencies in naive set comprehension principles. To address these issues and provide a rigorous basis for Cantor's transfinite numbers and his own 1904 proof of the well-ordering theorem, Ernst Zermelo developed the first axiomatic system for set theory. In his 1904 paper, "A New Proof of the Possibility of a Well-Ordering," Zermelo implicitly assumed the existence of the empty set while employing the axiom of choice to demonstrate that every set can be well-ordered, but he did not yet provide an explicit axiomatization. Zermelo's seminal 1908 paper, "Untersuchungen über die Grundlagen der Mengenlehre I," marked the explicit introduction of the axiom as part of his foundational framework. This , designated as Axiom II (the Axiom of Elementary Sets), asserts the existence of a set containing no elements whatsoever, which Zermelo denoted as "0." By including this , Zermelo replaced the problematic unrestricted comprehension principle with bounded separation and ensured a concrete starting point for set construction, thereby avoiding the foundational ambiguities exposed by . Zermelo emphasized the empty set's crucial role as the initial building block in the iterative hierarchy of sets, from which all other sets could be generated through operations like and union. He described this process: "We shall call these sets the basic sets or the sets of the first, second, third, ... type, according as they are formed in one, two, three, ... steps from the ." This approach provided a structured for , serving as the basis for deriving natural numbers and higher structures without relying on circular definitions or infinite descending membership chains.

Evolution in Von Neumann-Bernays-Gödel Set Theory

In John von Neumann's 1925 axiomatization of , the plays a foundational role in the construction of ordinal numbers, where it is explicitly identified as the least ordinal, denoted as , with subsequent ordinals built cumulatively upon it, such as 1 = {∅} and 2 = {∅, {∅}}. This approach integrates the directly into the hierarchy of ordinals without a separate existence axiom, deriving its properties from broader principles of set formation and well-ordering within the system. The Von Neumann-Bernays-Gödel (NBG) system, developed in the 1930s, extends this framework to a theory of both sets and proper classes, where the axiom of the empty set is explicitly stated to ensure the existence of an empty set within the domain of sets: there exists a set SS such that SS is a set and for all XX, X∉SX \notin S. Paul Bernays formalized this in his 1937 presentation, adapting von Neumann's earlier ideas to distinguish sets from classes while maintaining the empty set's status as a foundational set object. Kurt Gödel further refined the system in 1940 for his consistency proof, incorporating the axiom to guarantee the empty set's availability for constructions in the cumulative hierarchy, even as proper classes handle collections too large to be sets. These developments in NBG axiomatization address the limitations of pure set theories by introducing proper classes, such as the universal class of all sets, which cannot be members of any class; the remains a set—eligible for membership in other sets and classes—but is not treated as a proper class itself, preserving its role in building the von Neumann hierarchy without risking paradoxes from class comprehension. This distinction ensures that while the exists universally as a set, the theory's class-level operations do not conflate it with non-set collections, facilitating rigorous handling of transfinite structures.

Interpretations and Implications

Philosophical and Ontological Perspectives

The ontological status of the empty set has been a subject of debate in the , centering on whether it constitutes a genuine abstract entity in a Platonic sense or serves merely as a convenient formal device without deeper existential import. In Platonic realism, the empty set is regarded as an independent mathematical object within the set-theoretic universe, existing timelessly and mind-independently as the unique set with no elements, foundational to constructing all other sets. This view aligns with the commitment to a rich ontology of sets implied by axiomatic systems like Zermelo-Fraenkel set theory, where the empty set's existence is postulated to ensure the coherence of the mathematical hierarchy. Critics, however, question this Platonistic commitment, arguing that the empty set might be a fictional construct justified only by its utility rather than intrinsic reality. For instance, E. J. Lowe has contended that the empty set lacks a well-defined identity criterion absent any members, rendering its ontological status problematic and suggesting that positing it does not necessitate the of an actual but rather a notional placeholder for logical convenience. Similarly, , in exploring set-theoretic foundations, defined the empty set via a no object satisfies (such as {x | x ≠ x}), integrating it into his of sets without elevating it to a primitive or specially privileged status, thereby treating it as derivable from broader logical principles rather than a standalone Platonic form. In metaphysical discussions, this positions the empty set not as a "universal set with no members" in the sense of encompassing all absences, but as a structured representation of sheer non-membership, distinct from mere nothingness yet potentially reducible to conceptual absence without positive ontological weight. From a constructivist perspective, such as , the poses no ontological quandary, as its existence follows directly from constructive methods: it is the collection for which no element can be exhibited, verifiable through the inability to produce a member, thus affirming it as a valid, mind-dependent without reliance on abstract . This contrasts sharply with classical , where the 's reality is assumed a priori as part of an objective mathematical realm, independent of construction or proof; intuitionists, by contrast, ground its legitimacy in explicit mental acts, avoiding what they see as the metaphysical extravagance of positing unconstructed abstracts.

Applications in Mathematics Beyond Set Theory

In topology, the empty set forms the empty topological space, which possesses the distinctive property of being both open and closed—termed clopen—in every . This openness stems from the empty set being the union of zero open sets, while its closedness follows from the complement of the empty set being the entire space, which is defined to be open. These properties ensure consistency in topological definitions, such as , where the empty set is compact as it admits the empty cover. In algebraic contexts, particularly , the serves as the initial object in the , denoted Set, where there exists a unique —the empty function—from the to any other set, facilitating universal constructions like products and coproducts. Additionally, within the power set of a set under the operation of union, the forms a where the acts as the , satisfying A∪∅=AA \cup \emptyset = A for any AA, which underpins algebraic operations on collections of subsets. In logic and , the represents the empty in theory, a set containing no strings, which differs from the singleton {ϵ}\{\epsilon\} consisting solely of the empty string ϵ\epsilon; notably, the operation yields ∅∗={ϵ}\emptyset^* = \{\epsilon\}, highlighting its foundational role in closure properties. In relational databases, the corresponds to the empty relation, embodying a query result with no tuples, such as an SQL SELECT statement on an empty table returning zero rows to indicate falsehood, ensuring precise handling of absent data in .

References

  1. https://plato.stanford.edu/entries/set-theory/zf.html
  2. https://plato.stanford.edu/entries/zermelo-set-theory/
  3. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
  4. https://people.math.ethz.ch/~halorenz/4students/LogikGT/Ch13.pdf
  5. https://mathworld.wolfram.com/AxiomoftheEmptySet.html
  6. https://scholarworks.boisestate.edu/cgi/viewcontent.cgi?article=1000&context=math_undergraduate_theses
  7. https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Lian.pdf
  8. https://intuitionistic.files.wordpress.com/2010/07/nordstrom-et-al-type-theory.pdf
  9. https://ncatlab.org/nlab/show/cumulative+hierarchy
  10. https://eudml.org/doc/149573
  11. http://philsci-archive.pitt.edu/4603/1/CantorVonNeumann-2011.pdf
  12. https://www.jstor.org/stable/2268862
  13. https://plato.stanford.edu/entries/platonism-mathematics/
  14. https://plato.stanford.edu/entries/set-theory/
  15. https://plato.stanford.edu/entries/nothingness/
  16. https://philosophy-science-humanities-controversies.com/listview-details.php?id=243268&a=%24a&first_name=W.V.O.&author=Quine&concept=Empty%2520Set
  17. https://andrewmbailey.com/pvi/WhatIsOntologicalCategory.pdf
  18. https://iep.utm.edu/intuitionism-math/
  19. https://plato.stanford.edu/entries/set-theory-constructive/
  20. https://math.mit.edu/~hrm/palestine/munkres-topology.pdf
  21. http://files.farka.eu/pub/Awodey_S._Category_Theory%28en%29%28305s%29.pdf
  22. http://ozark.hendrix.edu/~yorgey/pub/monoid-pearl.pdf
  23. https://drive.uqu.edu.sa/_/mskhayat/files/MySubjects/20189FS%2520ComputationTheory/Introduction%2520to%2520the%2520theory%2520of%2520computation_third%2520edition%2520-%2520Michael%2520Sipser.pdf
  24. https://www.oreilly.com/library/view/sql-and-relational/9781449319724/ch12s07.html
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