Math 2.0

Mathematical logic and foundations Definition

Universal class

Builds on SignatureStructureSentenceUniversal quantifier

Definition

Let σ\sigma be a first-order signature. A universal sentence of σ\sigma is a sentence of the form ∀x1⋯∀xn φ\forall x_1 \cdots \forall x_n \, \varphi, with n≥0n \ge 0 universal quantifiers. Here φ\varphi is a quantifier-free formula whose free variables are among x1,…,xnx_1, \ldots, x_n. Let KK be a class of σ\sigma-structures. Then KK is a universal class if there is a set Σ\Sigma of universal sentences of σ\sigma whose models are exactly the members of KK.

Remarks

Example: simple graphs, in the signature with one binary relation EE, form a universal class with the axioms ∀x ¬E(x,x)\forall x \, \neg E(x, x) and ∀x∀y (E(x,y)→E(y,x))\forall x \forall y \, (E(x, y) \rightarrow E(y, x)).

Non-example: fields, in the signature (+,−,⋅,0,1)(+, -, \cdot, 0, 1), do not form a universal class. A universal sentence true in a structure is true in each substructure, because it only speaks about tuples of elements and the operations on them. The ring ℤ\Z is a substructure of the field ℚ\Q in this signature, and it is not a field.

Every universal class is an elementary class. The Łoś–Tarski preservation theorem gives the converse characterization: an elementary class is universal exactly when it is closed under substructures.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Sentence, concept, no evidence levelSentenceconceptSignature, concept, no evidence levelSignatureconceptStructure, concept, no evidence levelStructureconceptUniversal quantifier, concept, no evidence levelUniversal quantifierconceptUniversal class, definition, no evidence levelUniversal classdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Sentence concept
  2. Signature concept
  3. Structure concept
  4. Universal quantifier concept
  5. Universal class definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Elementary_class
About this page
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MSC03C52, 03C40
DefinesUniversal class
UsesSignature, Structure, Sentence, Universal quantifier
Changed2 November 2026

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