• README ⤓ .txt 2026-07-17T121634.146 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 2026-07-16T165328.000 000000000041552
  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.
• Elementary Geometry ⤓ .md 2026-07-16T165328.000 000000000053168
  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.
• 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 2026-07-16T145952.000 000000000050448
  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 2026-07-16T155351.000 000000000052424
  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 2026-07-16T163059.000 000000000050840
  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.