• README ⤓ .txt 2026-10-22T165042.936 000000000000760
  Axiomatizations of order, algebra, and geometry: theories whose signature governs a structure rather than a foundation. Dense linear orders, algebraically closed fields, real closed fields, Tarski's elementary geometry, and the theories of groups, rings, and fields are first-order theories in the same sense Peano arithmetic is — a signature, axioms, classical first-order consequence — and they are the theories the model theory in Metatheory is about.
• Algebraically Closed Fields ⤓ .md 2026-07-16T145804.000 000000000050040
  Tarski's quantifier elimination for algebraically closed fields (1930s; published in A Decision Method for Elementary Algebra and Geometry, 1948/1951) established completeness and decidability of ACF_p. Robinson recast it as model completeness and used it to give a model-theoretic proof of Hilbert's Nullstellensatz. Morley's categoricity theorem (1965) has ACF as its motivating example; the stability-theoretic analysis of ACF as a strongly minimal theory is the origin of geometric model theory.
• Dense Linear Orders ⤓ .md 2026-08-20T022144.000 000000000052960
  Cantor (1895) proved that any two countable dense linear orders without endpoints are isomorphic — the back-and-forth argument, and the first categoricity theorem in mathematics. Langford (1927) gave the quantifier elimination and the decision procedure, making DLO the first nontrivial theory shown complete and decidable, two years before Presburger's arithmetic.
• Differentially Closed Fields ⤓ .md 2026-10-22T180154.561 000000000000928
  Abraham Robinson, "On the concept of a differentially closed field" (1959), defined the class as the existentially closed differential fields and proved its theory complete, model complete, and decidable, axiomatizing it through the Ritt–Kolchin theory of differential ideals. Lenore Blum's MIT thesis (Generalized Algebraic Structures: A Model Theoretical Approach, 1968; published as "Differentially closed fields: a model-theoretic tour" in Contributions to Algebra, 1977) replaced Robinson's axioms with a schema on one-variable differential polynomials and proved the theory ω-stable. Saharon Shelah, "Differentially closed fields" (Israel Journal of Mathematics 16, 1973), proved uniqueness of the differential closure; Ellis Kolchin, Maxwell Rosenlicht, and Shelah independently (1973–74) proved its non-minimality. Ehud Hrushovski, "The Mordell–Lang conjecture for function fields" (Journal of the American Mathematical Society 9, 1996), proved the conjecture through the stability theory of DCF₀ and of separably closed fields.
• Elementary Geometry ⤓ .md 2026-10-22T165042.936 000000000000872
  Tarski lectured on the system at Warsaw in 1926–27; publication was delayed, first by other projects and then by the war, which destroyed the galley proofs. The axioms appeared in 1948, and a reduced set in "What is elementary geometry?" (in Henkin, Suppes, and Tarski, eds., The Axiomatic Method, 1959). Gupta's Berkeley thesis (1965) removed redundancies; the definitive treatment is Schwabhäuser, Szmielew, and Tarski, Metamathematische Methoden in der Geometrie (1983). Tarski and Givant, "Tarski's system of geometry" (BSL 5, 1999), give the history.
• P-adic Fields ⤓ .md 2026-10-22T180154.561 000000000000808
  James Ax and Simon Kochen, "Diophantine problems over local fields I–II" (American Journal of Mathematics 87, 1965; part II subtitled "A complete set of axioms for p-adic number theory") and "Diophantine problems over local fields III. Decidable fields" (Annals of Mathematics 83, 1966), and Yuri Eršov independently ("On the elementary theory of maximal normed fields", Doklady Akademii Nauk SSSR 165, 1965, with sequels in Algebra i Logika), axiomatized the first-order theory of ℚ_p and proved it decidable. Angus Macintyre, "On definable subsets of p-adic fields" (Journal of Symbolic Logic 41, 1976), proved quantifier elimination in the language with the power predicates. Alexander Prestel and Peter Roquette, Formally p-adic Fields (Lecture Notes in Mathematics 1050, 1984), generalized the theory to p-valued fields of arbitrary fixed p-rank. Jan Denef proved Serre's rationality conjecture for Poincaré series (Inventiones Mathematicae 77, 1984) and the cell decomposition theorem ("p-adic semi-algebraic sets and cell decomposition", Journal für die reine und angewandte Mathematik 369, 1986).
• Pseudofinite Fields ⤓ .md 2026-10-22T180154.561 000000000000856
  James Ax, "The elementary theory of finite fields" (Annals of Mathematics 88, 1968, pp. 239–271), proved the theory of all finite fields decidable, axiomatized its infinite models, and classified its completions. The arithmetic input is Serge Lang and André Weil, "Number of points of varieties in finite fields" (American Journal of Mathematics 76, 1954, pp. 17–20) — the Riemann hypothesis for curves in estimate form. The term quasi-finite is Serre's, for a perfect field with absolute Galois group Ẑ (Local Fields); the term pseudo-algebraically closed appears in the title of Gerhard Frey, "Pseudo algebraically closed fields with non-Archimedean real valuations" (Journal of Algebra 26, 1973), and Michael Fried and Moshe Jarden, Field Arithmetic (Ergebnisse der Mathematik, Springer), is the reference for the PAC generalization. Zoé Chatzidakis, Lou van den Dries, and Angus Macintyre, "Definable sets over finite fields" (Journal für die reine und angewandte Mathematik 427, 1992, pp. 107–135), proved the counting theorem that gives every definable set a dimension and a measure; Dugald Macpherson and Charles Steinhorn, "One-dimensional asymptotic classes of finite structures" (Transactions of the American Mathematical Society 360, 2008, pp. 411–448), turned that theorem into a definition.
• Random Graph ⤓ .md 2026-10-22T180154.561 000000000000800
  Wilhelm Ackermann's "Die Widerspruchsfreiheit der allgemeinen Mengenlehre" (1937) constructed the graph first, on the hereditarily finite sets — the same paper that codes V_ω into ℕ (see Hereditarily Finite Set Theory.md). Paul Erdős and Alfréd Rényi, "Asymmetric graphs" (Acta Mathematica Academiae Scientiarum Hungaricae 14, 1963), proved that almost all finite graphs are rigid and, as the contrast case, that the countable random graph almost surely has infinitely many automorphisms — an argument that also makes it one graph up to isomorphism, a uniqueness the paper leaves implicit. Richard Rado, "Universal graphs and universal functions" (Acta Arithmetica 9, 1964), gave the explicit universal graph on ℕ. Haim Gaifman (1964) proved the extension axioms complete; Ronald Fagin, "Probabilities on finite models" (Journal of Symbolic Logic, 1976), tied the theory to the asymptotics of finite graphs.
• Real Closed Fields ⤓ .md 2026-07-16T145838.000 000000000049512
  Artin and Schreier (1927) gave the algebraic theory of real closed fields and used it to solve Hilbert's seventeenth problem. Tarski proved quantifier elimination, completeness, and decidability for the ordered field of reals — work of the early 1930s, delayed by the war, published as A Decision Method for Elementary Algebra and Geometry (1948, revised 1951). Collins (1975) gave cylindrical algebraic decomposition, the first implementable procedure. Van den Dries, Pillay, and Steinhorn founded o-minimality (1980s) with RCF as the motivating case.
• Theories of Algebraic Structures ⤓ .md 2026-08-20T065417.000 000000000056720
  The axiomatizations are nineteenth-century mathematics; their status as logical objects dates from Tarski, Mostowski, and Robinson's Undecidable Theories (1953), which proved group theory, ring theory, lattice theory, and field theory undecidable by interpreting Q in each. Szmielew ("Elementary properties of Abelian groups", 1955) proved the complementary result: the theory of abelian groups is decidable. Julia Robinson (1949) showed ℤ is definable in ℚ, hence Th(ℚ) is undecidable; Ax and Kochen (1965) gave the transfer principle for p-adic fields.
• Theory of Boolean Algebras ⤓ .md 2026-10-22T090044.000 000000000058496
  Tarski found the decidability of the elementary theory in 1940 and announced it in 1949 — the same abstract in which he announced ACF, RCF, and elementary geometry. Skolem had given quantifier elimination for the theory with an added ideal predicate. The elementary classification is Ershov's ("Decidability of the elementary theory of distributive lattices with relative complements and the theory of filters", 1964) and, independently, Keisler's. Tarski's own invariants for the countable case predate them.
• Well-Orderings ⤓ .md 2026-10-22T161354.000 000000000057336
  Tarski's first paper on the subject is from 1921 ("Przyczynek do aksjomatyki zbioru dobrze uporządkowanego"). Mostowski and Tarski announced the elementary classification in 1949 ("Arithmetical classes and types of well-ordered systems"); the full analysis waited nearly thirty years and appeared as Doner, Mostowski, and Tarski, "The elementary theory of well-ordering — a metamathematical study" (Logic Colloquium '77, 1978, pp. 1–54). Doner and Tarski, "An extended arithmetic of ordinal numbers" (Fund. Math. 65, 1969), handles the ordinal operations. Jeřábek (2024) gives a short proof of both the axiomatization and the decidability.
• CRITERIA ⤓ .txt 2026-07-16T145702.000 000000000018928
  Not sufficient: Being a first-order theory, since every entry in this division is one. Being applied to a structure. A theory whose interest is what it can prove about itself or about other theories, rather than what satisfies it, belongs with the foundational groups above: arithmetic entries axiomatize number and its fragments, set entries membership, truth entries a truth predicate, part entries parthood. The test is what the entry is measured by — strength, or models.