Tarski–Grothendieck set theory

Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom which states that for each set there is a Grothendieck universe it belongs to (see below). Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than that of conventional set theories such as ZFC. For example, adding this axiom supports category theory.

The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs.

Axioms

Tarski–Grothendieck set theory starts with conventional Zermelo–Fraenkel set theory and then adds "Tarski's axiom". We will use the axioms, definitions, and notation of Mizar to describe it. Mizar's basic objects and processes are fully formal; they are described informally below. First, let us assume that:

Given any set $A$ , the singleton $\{A\}$ exists.
Given any two sets, their unordered and ordered pairs exist.
Given any family of sets, its union exists.

TG includes the following axioms, which are conventional because they are also part of ZFC:

Set axiom: Quantified variables range over sets alone; everything is a set (the same ontology as ZFC).
Extensionality axiom: Two sets are identical if they have the same members.
Axiom of regularity: No set is a member of itself, and circular chains of membership are impossible.
Axiom schema of replacement: Let the domain of the function $F$ be the set $A$ . Then the range of $F$ (the values of $F(x)$ for all members $x$ of $A$ ) is also a set.

It is Tarski's axiom that distinguishes TG from other axiomatic set theories. Tarski's axiom also implies the axioms of infinity, choice,^[1]^[2] and power set.^[3]^[4] It also implies the existence of inaccessible cardinals, thanks to which the ontology of TG is much richer than that of conventional set theories such as ZFC.

Tarski's axiom (adapted from Tarski 1939^[5]). For every set $x$ , there exists a set $y$ whose members include:

- $x$ itself;

- every subset of every member of $y$ ;

- the power set of every member of $y$ ;

- every subset of $y$ of cardinality less than that of $y$ .

More formally:

\forall x\exists y[x\in y\wedge \forall z\in y({\mathcal {P}}(z)\subseteq y\wedge {\mathcal {P}}(z)\in y)\wedge \forall z\in {\mathcal {P}}(y)(\neg z\approx y\to z\in y)]

where " ${\mathcal {P}}(x)$ " denotes the power class of x and " $\approx$ " denotes equinumerosity. What Tarski's axiom states (in the vernacular) for each set $x$ there is a Grothendieck universe it belongs to.

Implementation in the Mizar system

The Mizar language, underlying the implementation of TG and providing its logical syntax, is typed and the types are assumed to be non-empty. Hence, the theory is implicitly taken to be non-empty. The existence axioms, e.g. the existence of the unordered pair, is also implemented indirectly by the definition of term constructors.

The system includes equality, the membership predicate and the following standard definitions:

Singleton: A set with one member;
Unordered pair: A set with two distinct members. $\{a,b\}=\{b,a\}$ ;
Ordered pair: The set $\{\{a,b\},\{a\}\}=(a,b)\neq (b,a)$ ;
Subset: A set all of whose members are members of another given set;
The union of a family of sets $Y$ : The set of all members of any member of $Y$ .

Implementation in Metamath

The Metamath system supports arbitrary higher-order logics, but it is typically used with the "set.mm" definitions of axioms. The ax-groth axiom adds Tarski's axiom, which in Metamath is defined as follows:

⊢ ∃y(x ∈ y ∧ ∀z ∈ y (∀w(w ⊆ z → w ∈ y) ∧ ∃w ∈ y ∀v(v ⊆ z → v ∈ w)) ∧ ∀z(z ⊆ y → (z ≈ y ∨ z ∈ y)))

Notes

↑ Tarski (1938)
↑ http://mmlquery.mizar.org/mml/current/wellord2.html#T26
↑ Robert Solovay, Re: AC and strongly inaccessible cardinals.
↑ Metamath grothpw.
↑ Tarski (1939)

References

Andreas Blass, I.M. Dimitriou, and Benedikt Löwe (2007) "Inaccessible Cardinals without the Axiom of Choice," Fundamenta Mathematicae 194: 179-89.
Bourbaki, Nicolas (1972). "Univers". In Michael Artin; Alexandre Grothendieck; Jean-Louis Verdier. Séminaire de Géométrie Algébrique du Bois Marie – 1963-64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – vol. 1 (Lecture notes in mathematics 269) (in French). Berlin; New York: Springer-Verlag. pp. 185–217.
Patrick Suppes (1960) Axiomatic Set Theory. Van Nostrand. Dover reprint, 1972.
Tarski, Alfred (1938). "Über unerreichbare Kardinalzahlen" (PDF). Fundamenta Mathematicae. 30: 68–89.
Tarski, Alfred (1939). "On the well-ordered subsets of any set" (PDF). Fundamenta Mathematicae. 32: 176–183.

External links

Trybulec, Andrzej, 1989, "Tarski–Grothendieck Set Theory", Journal of Formalized Mathematics.
Metamath: "Proof Explorer Home Page." Scroll down to "Grothendieck's Axiom."
PlanetMath: "Tarski's Axiom"

Set theory

Axioms	Choice countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification

Operations	Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union

Concepts Methods	Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram

Set types	Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset · Superset Transitive Uncountable Universal

Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck

Paradoxes Problems	Russell's paradox Suslin's problem

Set theorists	Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine

Mathematical logic

General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory

Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic

Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Naive set theory	Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image Map Function Relation Ordered pair

Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

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