Mathematical structures + their formal-language descriptions: Tarski semantics, completeness/compactness, Łoś ultraproduct, types, stability theory (Shelah), o-minimality, model-theoretic algebraic-geometry.
model-theory
Tarski semantic truth
Tarski 1933/1944: 'snow is white' is true iff snow is white. Formal-truth-definition for fixed object-language in metalanguage. Foundation…
Compactness theorem (Gödel)
Gödel 1930: set of first-order sentences satisfiable iff every finite-subset is. Equivalent to completeness. Foundational…
Łoś ultraproduct theorem
Łoś 1955: ultraproduct of structures satisfies sentence iff ultrafilter-many factors do. Constructs nonstandard-models / hyperreals…
Types + saturation / omitting
Type p(x) over A = consistent set of formulas. ω-saturated model realises all types. Omitting-types theorem: countable theory has model…
Shelah stability theory
Shelah 1971-90: classify first-order theories by # of models in each cardinality. Stable/superstable/ω-stable.…
o-minimality (Pillay-Steinhorn)
Pillay-Steinhorn 1986: ordered-structure where every definable-set is finite-union of points + intervals. Real-closed-fields,…
Model theory of fields
ACF algebraically-closed fields: ℵ_α-categorical for α>0; quantifier-elimination → Chevalley constructible-sets. RCF real-closed: Tarski…
NIP theory (Shelah)
Shelah 1971: theory has independence-property (IP) iff some formula has VC-dim infinity. NIP theories: bounded-VC. Includes ACF + RCF +…
Definable sets + quantifier elimination
Theory T admits QE iff every formula equivalent to quantifier-free. Tarski-Seidenberg (RCF) / ACF / Presburger (linear arithmetic over ℤ).…
Nonstandard analysis (Robinson)
Robinson 1966: hyperreals *ℝ ⊃ ℝ contain infinitesimals + infinities. Rigorous foundation for Leibniz-style infinitesimal-calculus.…
Morley categoricity theorem
Morley 1965: countable first-order T categorical in some uncountable cardinality is categorical in all. Total-transcendence-rank notion.…
Zilber-Hrushovski geometry of strongly-minimal sets
Zilber 1980s trichotomy conjecture; Hrushovski 1988 counter-example. Strongly-minimal sets fall into geometric classes (trivial /…
Compactness theorem (Godel-Malcev)
Godel 1930 / Malcev 1936: a set of first-order sentences has a model iff every finite subset has a model; foundation of model theory +…
Lowenheim-Skolem theorem
Lowenheim 1915 / Skolem 1920: countable theory has model of any infinite cardinality kappa; downward + upward versions; reveals first-order…
Morley categoricity theorem
Morley 1965: a complete theory T countable in L is categorical in some uncountable cardinality iff categorical in all uncountable…
Ultraproduct construction (Los theorem)
Los 1955: ultraproduct of structures over ultrafilter satisfies same first-order sentences as 'most' factors; basis of non-standard models…
O-minimal structures (Pillay-Steinhorn)
Pillay-Steinhorn 1988: ordered structures where every definable set is finite union of points + intervals; tame topology of definable sets;…
Zariski-Noetherian irreducibility (algebraic closure)
Algebraic-closed-field theory ACF_p categorical in uncountable cardinalities; Zariski-topology Noetherian; each variety uniquely decomposes…