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Nice Properties of Theories

 
ω-stable
ω-stable
    A theory is \(\omega\)-stable if \(S^M_n(A)\) is countable for all models \(M\) and countable subsets \(A\subseteq M\).

    See , , .
    superstable
    superstable
      Given an infinite cardinal \(\kappa\), a theory is \(\kappa\)-stable if \[|S_n^M(A)|\leq\kappa\] for all models \(M\) and subsets \(A\subseteq M\) with \(|A|\leq\kappa\).

      A theory \(T\) is superstable if there is some cardinal \(\lambda\) such that \(T\) is \(\kappa\)-stable for all \(\kappa\geq\lambda\).

      See , , .

      Remarks
      In the definition, \(\lambda\) can be made equal to \(2^{|T|}\). Some sources define superstable as "stable and supersimple".
      stable (NOP)
      stable (NOP)
        Given an infinite cardinal \(\kappa\), a theory is \(\kappa\)-stable if \[|S_n^M(A)|\leq\kappa\] for all models \(M\) and subsets \(A\subseteq M\) with \(|A|\leq\kappa\).

        A theory is stable if it is \(\kappa\)-stable for some infinite cardinal \(\kappa\).

        See , , .

        Alternate definition:
        A formula \(\varphi(x,y)\) has the order property (OP) if there are \((a_i)_{i\lt\omega}\) and \((b_i)_{i\lt\omega}\) such that \[\models\varphi(a_i,b_j)~\Leftrightarrow~ i\lt j.\] A theory is stable (NOP) if no formula has the order property.

        See .
        o-minimal
        o-minimal
          Assume the language \(\mathcal{L}\) contains a binary relation <, and that T is an \(\mathcal{L}\)-theory in which < is a linear order. T is o-minimal if, given a model M, every definable subset of M can be written as a finite union of points in M and intervals with endpoints in M.

          characterization of forking

          See , , .
          dp-minimal
          dp-minimal
            A theory has dp-rankn if there are formulas \(\varphi_1(x,y),\ldots,\varphi_n(x,y)\) and mutually indiscernible sequences \((a^1_i)_{i\lt\omega},\ldots,(a^n_i)_{i\lt\omega}\) such that for any function \(\sigma:\{1,\ldots,n\}\longrightarrow\omega\), the type \[\{\varphi_k(x,a^k_{\sigma(k)}):k\leq n\}\cup\] \[\{\neg\varphi_k(x,a^k_i):i\neq\sigma(k),~k\leq n\}\] is consistent.

            A theory is dp-minimal if it has dp-rank 1.

            See , .
            NIP
            NIP (dependent)
              A formula \(\varphi(x,y)\) as the independence property (IP) if there are \((a_i)_{i\lt\omega}\) and \((b_I)_{I\subseteq\omega}\) such that \[\models\varphi(a_i,b_I)~\Leftrightarrow~i\in I.\] A theory is NIP (dependent) if no formula has the independence property.

              See , .
              supersimple
              supersimple
                A theory is supersimple if for all sets \(B\) and complete types \(p\in S_n(B)\) there is a finite subset \(A\subseteq B\) such that \(p\) does not fork over \(A\).

                See , .
                simple (NTP)
                simple (NTP)
                  A theory \(T\) is simple if for all sets \(B\) and complete types \(p\in S_n(B)\) there is a subset \(A\subseteq B\), with \(|A|\leq|T|\), such that \(p\) does not fork over \(A\).

                  Alternate Definition:
                  A formula \(\varphi(x,y)\) has the tree property (TP) if there are \((a_\eta:\eta~\in\omega^{\lt\omega})\) and some \(k\geq 2\) such that
                  • \(\forall~\sigma\in\omega^\omega\), \(\{\varphi(x,a_{\sigma|_n}):n\lt\omega\}\) is consistent.
                  • \(\forall~\eta\in\omega^{\lt\omega}\), \(\{\varphi(x,a_{\sigma\hat{~}n}:n\lt\omega\}\) is \(k\)-inconsistent.
                  A theory is simple (NTP) if no formula has the tree property.

                  See , .

                  Remarks:
                  One often studies theories in which forking independence is well-behaved. Simple theories have been shown to strongly exemplify this. In particular, the following are equivalent (see , ):
                  • T is simple.
                  • Nonforking (or nondividing) is symmetric.
                  • Nonforking (or nondividing) is transitive.
                  • Nonforking (or nondividing) has local character.
                  • There is a notion of independence (which must be nonforking) satisfying certain properties (see ).
                  NSOP1
                  NSOP1
                  no proper examples known
                    A formula \(\varphi(x,y)\) has SOP1 if there are \((a_\eta:\eta~\in 2^{\lt\omega})\) such that
                    • \(\forall~\sigma\in 2^\omega\), \(\{\varphi(x,a_{\sigma|_n}):n\lt\omega\}\) is consistent.
                    • \(\forall~\eta,\mu\in\omega^{\lt\omega}\), if \(\mu\hat{~}0\prec\eta\), then \(\{\varphi(x,a_{\mu\hat{~}1}),~\varphi(x,a_{\eta})\}\) is inconsistent.
                    A theory is NSOP1 if no formula has SOP1.

                    See , .
                    NTP1
                    NTP1 (NSOP2)
                    no proper examples known
                      A formula \(\varphi(x,y)\) has the tree property 1 (TP1) if there are \((a_\eta:\eta~\in \omega^{\lt\omega})\) such that
                      • \(\forall~\sigma\in \omega^\omega\), \(\{\varphi(x,a_{\sigma|_n}):n\lt\omega\}\) is consistent.
                      • \(\forall\) incomparable \(\eta,\mu\in\omega^{\lt\omega}\), \(\{\varphi(x,a_\mu),~\varphi(x,a_{\eta})\}\) is inconsistent.
                      A theory is NTP1 if no formula has TP1.

                      See , .

                      Remarks
                      As a property of the theory, TP1 is equivalent to SOP2 (see ), which was recently shown to be maximal in Keisler's order (see ).
                      NSOP3
                      NSOP3
                      no proper examples known
                        For \(n\geq 3\), a formula \(\varphi(x,y)\) has the n-strong order property (SOPn) if there are \((a_i)_{i\lt\omega}\) such that \(\models\varphi(a_i,a_j)\) for all \( i \lt j \); but \[\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}\] is inconsistent.

                        A theory is NSOPn if no formula has SOPn.

                        See , .
                        NSOP4
                        NSOP4
                          For \(n\geq 3\), a formula \(\varphi(x,y)\) has the n-strong order property (SOPn) if there are \((a_i)_{i\lt\omega}\) such that \(\models\varphi(a_i,a_j)\) for all \( i\lt j \); but \[\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}\] is inconsistent.

                          A theory is NSOPn if no formula has SOPn.

                          See , , .

                          Remarks:
                          There are results about non-existence of universal models of cardinality λ (for certain λ) when T has SOP4 (see , ).
                          For example:
                          If T has SOP4 and λ is regular such that κ+ < λ < 2κ for some κ, then T has no universal model of cardinality λ.
                          NSOPn+1
                          NSOPn+1
                            For \(n\geq 3\), a formula \(\varphi(x,y)\) has the n-strong order property (SOPn) if there are \((a_i)_{i\lt\omega}\) such that \(\models\varphi(a_i,a_j)\) for all \( i\lt j \); but \[\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}\] is inconsistent.

                            A theory is NSOPn if no formula has SOPn.

                            See , .
                            NSOP
                            NSOP
                              A formula \(\varphi(x,y)\) has the fully finitary strong order property (SOP) if there are \((a_i)_{i\lt\omega}\) such that \(\models\varphi(a_i,a_j)\) for all \( i\lt j \); but, for all \(n\geq 3\), \[\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}\] is unsatisfiable.

                              A theory is NSOP if no formula has SOP.

                              See , .

                              Remark: SOP is a not currently the standard abbreviation.

                              Two other dividing lines with a close relationship to the fully finitary strong order property are the finitary strong order property and the strong order property. The definitions of these properties are obtained by replacing the formula \(\varphi(x,y)\) in the definition above with, respectively, a type in finitely many variables and a type in arbitrarily many variables. Note that compactness does not imply these are the same, due to the requirement that the type omit n-cycles for all n.
                              NSOP
                              NSOP
                                A formula \(\varphi(x,y)\) has the strict order property (SOP) if there are \((a_i)_{i\lt\omega}\) such that \[\models\exists x(\varphi(x,a_j)\wedge\neg\varphi(x,a_i))~\Leftrightarrow~i\lt j.\] A theory is NSOP if no formula has the strict order property.

                                See , .

                                NTP2
                                NTP2
                                  A formula \(\varphi(x,y)\) has the tree property 2 (TP2) if there are \((a_{i,j})_{i,j\lt\omega}\) such that
                                  • \(\forall~\sigma\in\omega^\omega\), \(\{\varphi(x,a_{n,\sigma(n)}):n\lt\omega\}\) is consistent.
                                  • \(\forall~n\lt\omega,~\forall~ i\lt j\lt\omega\), \(\{\varphi(x,a_{n,i}),~\varphi(x,a_{n,j})\}\) is inconsistent.
                                  A theory is NTP2 if no formula has the tree property 2.

                                  See , .
                                  forking = dividing
                                    nonforking exists
                                      strongly minimal
                                      strongly minimal
                                        A theory is strongly minimal if for all models M, any definable subset of M is finite or cofinite.

                                        See , .

                                        Remarks
                                        For simplicity, this region of the map is for strongly minimal theories in a countable language. With this caveat, a strongly minimal theory is \(\omega\)-stable, as indicated by the map.
                                        Click a property above to highlight region and display details. Or click the map for specific region information.
                                        Reset
                                        List of Examples
                                          Implications Between Properties
                                          Click the link to see the implications between all the above properties:
                                          Diagram of Implications

                                          Some other important equivalences:
                                          • stable ⇔ NIP and NSOP
                                          • simple ⇔ NTP1 and NTP2
                                          For proofs of these implications, see , , , , , , .
                                          Some of the proofs have been collected here.
                                          Open Regions
                                          Each small region outlined in red currently contains no known examples, due to the following open questions:
                                          • Are the implications, SOP3 ⇒ TP1 ⇒ SOP1, proper? See , , .
                                          • Is the implication, SOP1 ⇒ TP, proper? The answer to this is thought to be yes, witnessed by Tfeq, but a full proof has not been given. See the comments section of Tfeq.
                                          • Is the SOP hierarchy strict inside of NTP2? See , Exercise III.7.12.
                                          It is known that the {TP and NSOP3} region is nonempty (see the list of Open Examples). However, the {NTP2, NSOP, and TP} region may be empty.
                                          Open Examples
                                          The following examples are of well-known theories, whose positions on the map are open. Click an example for its definition, and to highlight the possible regions it could be in.
                                          • Tfeq
                                          • ω-free PAC fields
                                          • infinite-dimensional vector space with a bilinear form
                                          NIP and NSOP
                                          NIP and SOP
                                          IP and NSOP
                                          IP and SOP
                                          infinite sets
                                          E.g. take the theory of \(\mathbb{N}\) in the empty language.

                                          characterization of forking
                                          This theory has quantifier elimination, so definable sets (in one variable) are finite or cofinite.

                                          See .
                                          ACF
                                          algebraically closed fields

                                          Axiomatized by finitely many field axioms (in \(\mathcal{L}=\{+,\cdot,0,1\}\)), and the infinite axiom scheme saying every polynomial has a root.

                                          characterization of forking
                                          The theory can be completed by specifying the characteristic. By quantifier elimination, definable sets (in one variable) are Boolean combinations of polynomial zerosets and so finite or cofinite.

                                          See .
                                          \(\mathbb{Q}\)-vector spaces
                                          Axiomatized by the axioms for torsion-free divisible abelian groups in \(\mathcal{L}=\{+,0\}\).

                                          characterization of forking
                                          By quantifier elimination, definable sets (in one variable) are given as Boolean combinations of formulas of the form nx = a, where n is an integer and a is an arbitrary element. Since the group is torsion free, such sets are finite or cofinite.

                                          See .
                                          DCF0
                                          differentially closed fields of characteristic 0

                                          characterization of forking
                                          Shown to be ω-stable by Blum.

                                          See .
                                          everywhere infinite forest
                                          Fraïssé limit of finite trees.

                                          characterization of forking
                                          Models consist of infinitely many infinitely branching trees. Not strongly minimal as the set of neighbors of a specific vertex is definable, infinite and coinfinite.

                                          Morley rank ω. Also referred to as the free pseudoplane.

                                          See , .
                                          \((\mathbb{Z},+,-,0,1)\)
                                          characterization of forking
                                          Quantifier elimination up to the definable predicates Pn, which distinguish the elements divisible by n, for n > 0. This shows superstability. Any set of prime numbers gives a complete type over \(\emptyset\) of an element divisible by exactly the prime numbers in the set. Thus \(S_1(\emptyset)\) is uncountable and the theory is not ω-stable.

                                          See .
                                          finitely refining equivalence relations
                                          Infinitely many equivalence relations \(\{E_0,E_1,\ldots\}\) such that \(E_0\) has two infinite classes, and each \(E_i\)-class is partitioned into two infinite \(E_{i+1}\) classes.
                                          See , .
                                          infinitely refining equivalence relations
                                          Infinitely many equivalence relations \(\{E_0,E_1,\ldots\}\) such that \(E_0\) has infinitely many infinite classes, and each \(E_i\)-class is partitioned into infinitely many infinite \(E_{i+1}\) classes.

                                          characterization of forking
                                          See , .
                                          DCFp
                                          differentially closed fields of characteristic p

                                          characterization of forking
                                          Shown to be stable, but not superstable, by Shelah (1973).

                                          See .
                                          free group on n > 1 generators
                                          Shown to be stable, but not superstable, by Sela (2006).

                                          See .
                                          RCF
                                          real closed fields

                                          Complete theory, with quantifier elimination in \(\mathcal{L}=\{+,\cdot,0,1,\lt\}\).
                                          See .
                                          \((\mathbb{Z},\lt)\)
                                          Theory of discrete linear orders.
                                          See .
                                          \((\mathbb{Q},\lt)\)
                                          Theory of dense linear orders.
                                          See .
                                          \((\mathbb{Z},+,-,\lt,0,1)\)
                                          Also known as Presburger Arithmetic.
                                          Quantifier elimination up to predicates Pn naming the elements divisible by n, for n > 0. Not o-minimal since, e.g., 2\(\mathbb{Z}\) is a definable infinite discrete set.

                                          See .
                                          \((\mathbb{Z},~x\mapsto x+1)\)
                                          characterization of forking
                                          By quantifier elimination, definable sets (in one variable) are of Boolean combinations of formulas of the form x = a + n, where a is an arbitrary element and n is an integer. Since successors are unique, such sets are finite or cofinite.

                                          See .
                                          ACVF
                                          algebraically closed valued fields

                                          See .
                                          \((\mathbb{Q}_p,+,\cdot,v(x)\geq v(y))\)
                                          p-adic field with valuation
                                          See .
                                          random graph
                                          Axiomatized in graph language by sentences saying that if A and B are finite disjoint sets of vertices then there is a vertex connected to everything in A and nothing in B.

                                          characterization of forking
                                          Forking independence is the same as algebraic independence and algebraic closure is trivial, so the theory is supersimple. Not stable since xRy has the order property.

                                          See , .
                                          pseudo-finite fields
                                          Defined as ultraproducts of finite fields.
                                          See .
                                          ACFA
                                          algebraically closed fields with a generic automorphism

                                          characterization of forking
                                          See .
                                          QACFA
                                          quasi-algebraically closed fields with a generic automorphism

                                          See .
                                          ultraproduct of \(\mathbb{Q}_p\)
                                          \(\mathbb{Q}_p\) denotes the field of p-adic numbers
                                          See .
                                          generic Kn-free graph
                                          Given n ≥ 3, Kn denotes the complete graph on n vertices. Axiomatized in the graph language by sentences stating that if A and B are finite disjoint sets of vertices, with A Kn-1-free, then there is a vertex connected to everything in A and nothing in B.

                                          characterization of forking
                                          See , .
                                          \((\mathbb{Q},\textrm{cyc})\)
                                          The cyclic order on the rationals is cyc(a,b,c) if and only if a < b < c or b < c < a or c < a < b.
                                          Can be interpreted in a real closed field, so NIP. The unique 1-type over \(\emptyset\) proves "x = 0, or x = 1, or cyc(0,x,1), or cyc(1,x,0)". Each of these disjuncts divides over \(\emptyset\), so the type forks over \(\emptyset\), which shows that nonforking fails existence.

                                          See .
                                          SCFp
                                          n
                                          separably closed fields of characteristic p and Eršov invariant n ≤ ∞

                                          Given a separably closed field F of characteristic p, the Eršov invariant is \[n=[F:F^p]\in\mathbb{Z}^+\cup\{\infty\}.\]
                                          Shown to be stable by Macintyre, Shelah, and Wood (1975).

                                          See .
                                          \((\mathbb{R},+,\cdot,0,1,\textrm{exp})\)
                                          the real exponential field
                                          Shown to be o-minimal by Wilkie (1996).

                                          See .
                                          \(((\mathbb{Z}/4\mathbb{Z})^\omega,+)\)
                                          Totally categorical of Morley rank 2.

                                          See , .
                                          finitely cross-cutting equivalence relations
                                          Infinitely many equivalence relations \(\{E_0,E_1,\ldots\}\) such that each \(E_i\) has two infinite classes, and for all \(n\lt\omega\) and \(I\subseteq n\), \[\forall x\exists y\left(\bigwedge_{i\in I}E_i(x,y)\wedge\bigwedge_{i\in n\backslash I}\neg E_i(x,y)\right)\]
                                          See .
                                          universal bowtie-free graph
                                          Complete theory of the universal and existentially closed countable graph omitting a 'bowtie' (sum of two triangles sharing a single vertex).
                                          See , .
                                          universal graph omitting odd cycles of length ≤ n
                                          Given odd n ≥ 3, the complete theory of the universal and existentially closed countable graph omitting all odd cycles of length at most n.
                                          See , .
                                          universal directed graph omitting cycles of length ≤ n
                                          Complete theory of the universal and existentially closed countable directed graph omitting directed cycles of length ≤ n.
                                          See .
                                          ZFC
                                          Theory of set theory in the language \(\mathcal{L}=\{\in\}\).
                                          \((\mathbb{Z},+,\cdot,0,1)\)
                                          Complete theory of the ring of integers (a completion of Peano Arithmetic).
                                          \((\mathbb{Q}^n,\lt_1,\ldots,\lt_n)\)
                                          Complete theory of \(\mathbb{Q}^n\), for n > 1, with coordinate orderings. In particular, given \(x,y\in\mathbb{Q}^n\), set \[x\lt_i y ~\Leftrightarrow~ x_i\lt y_i. \]
                                          When n=1, this theory is the usual DLO. In general, the theory has dp-rank n.

                                          See .
                                          Urysohn sphere
                                          The unique universal and ultrahomogeneous separable metric space (with distances bounded by 1).

                                          characterization of forking
                                          There are few options for what language to use for this structure. One is to use distance relations \(d(x,y)\leq r\), for \(r\in\mathbb{Q}\). Alternatively, one consider the Urysohn sphere as a metric structure in continuous logic. See , .
                                          free nth root of the complete graph
                                          The unique universal and ω-homogeneous countable metric space with distances in {0,1,...,n}.

                                          characterization of forking
                                          This theory was originally defined with graph relations Ri for 0 ≤ in, where Ri(x,y) holds if and only if x and y can be connected by a path of length i. The graph structure and metric structure can be thought of as the same via
                                          d(x,y) = min{i : Ri(x,y)}

                                          See , .
                                          Hrushovski's new strongly minimal set
                                          Hrushovski constructed an example of a strongly minimal theory, which is not locally modular and does not interpret an infinite group. This disproved Zilber's conjecture that a strongly minimal theory must either be locally modular or interpret an infinite field.
                                          See , .
                                          infinitely cross-cutting equivalence relations
                                          Infinitely many equivalence relations \(\{E_0,E_1,\ldots\}\) such that each \(E_i\) has infinitely many infinite classes, and for all \(i\lt\omega\), each \(E_{i+1}\) class splits each \(E_i\) class into infinitely many pieces.
                                          See .
                                          non-simple generic limit of \((\mathcal{K}_f,\leq)\) for good \(f\)
                                          \((\mathcal{K}_f,\leq)\) is a class of finite structures (in a finite relational language), which is closed under free amalgamation and is equipped with a predimension and a control function \(f\).

                                          \(f\) is good if \(\mathcal{K}_f\) is closed under free \(\leq\)-amalgamation.

                                          We consider the theory of the generic structure \(\mathcal{M}_f\). These theories are always NSOP4, and the non-simple case is always SOP3 and TP2. The simple case can also be characterized by the closure of \(\mathcal{K}_f\) under independence theorem diagrams.
                                          See .
                                          VFA0
                                          The limit theory (as p→∞) of the Frobenius automorphism acting on an algebraically closed valued field of characteristic p.
                                          See .
                                          atomless Boolean algebras
                                          The theory of the Fraissé limit of Boolean algebras in the language \(\{0,1,\neg,\wedge,\vee\}\).
                                          See .
                                          generic Kr
                                          n
                                          -free r-graph
                                          Given n > r > 2, Kr
                                          n
                                          denotes the complete r-graph on n vertices.

                                          characterization of forking
                                          See .
                                          a strictly stable superflat graph
                                          Given integers m and n, let Km
                                          n
                                          be the class of graphs obtained from the complete graph on n vertices by replacing each edge with a path containing at most m new vertices.

                                          A graph G is superflat if for all m there is some n such that G omits Km
                                          n
                                          . G is ultraflat if there is some n such that for all m, G omits Km
                                          n
                                          .

                                          Define the following strictly stable superflat graph. Begin with vertex set ω∪ωω. For each σ in ωω and m < ω, add a path from σ to σ|m containing m new vertices.

                                          characterization of forking
                                          Any superflat graph is stable; any ultraflat graph (and so any planar graph) is superstable. The graph above interprets infinitely refining equivalence relations, and so is strictly stable.
                                          See , , .
                                          imperfect bounded PAC fields
                                          A variety defined over a field K is absolutely irreducible if it is not the union of two algebraic sets defined over some algebraically closed extension of K

                                          A field K is pseudo-algebraically closed (PAC) if every absolutely irreducible variety defined over K has a K-rational point.

                                          K is perfect if either it has characteristic 0, or if K has characteristic p then Kp = K. K is bounded if for all n > 1, K has finitely many separably algebraic extensions of degree n.

                                          characterization of forking
                                          If K is a PAC field then Th(K) is simple if and only if K is bounded. The theory of a perfect bounded PAC field is supersimple. See , , .
                                          Tfeq
                                          Consider the language \(\mathcal{L}=\{P,Q,E\}\), where \(P,Q\) are unary relations and \(E\) is a ternary relation. Tfeq is the model completion of the following theory:
                                          See , . In this theory is shown to be NSOP3 and not simple. In it is shown to be NTP1. In it is claimed to be NSOP1, but the proof is currently under scrutiny. See also .

                                          list of open examples
                                          ω-free PAC fields
                                          A variety defined over a field K is absolutely irreducible if it is not the union of two algebraic sets defined over some algebraically closed extension of K.

                                          A field K is pseudo-algebraically closed (PAC) if every absolutely irreducible variety defined over K has a K-rational point.

                                          A PAC field K is ω-free if there is an elementary substructure L of K, whose absolute Galois group Aut(Lsep/L) is isomorphic to \(\hat{F}_\omega\), which is defined below.

                                          Let \(F_\omega\) be the free group on countably many generators. Let \(\mathcal{N}\) be the family of normal, finite-index subgroups of \(F_\omega\) containing cofinitely many generators. Then \[\hat{F}_\omega=\varprojlim_{N\in\mathcal{N}}F_\omega/N\]
                                          characterization of forking
                                          See , . In this theory is shown to be NSOP3 and not simple; and remarked to probably be NSOP1. See also .

                                          list of open examples
                                          infinite-dimensional vector space with a bilinear form
                                          Defined in the 2-sorted language of vector spaces over an algebraically closed field.
                                          See , .

                                          list of open examples
                                          universal directed graph omitting all directed cycles
                                          Complete theory of the universal and existentially closed countable directed graph omitting directed cycles of length ≤ n.
                                          See .
                                          bounded pseudo real closed fields
                                          A field K, of characteristic 0, is bounded if, for every n, K has only finitely many extensions of degree n.

                                          A variety defined over a field K is absolutely irreducible if it is not the union of two algebraic sets defined over some algebraically closed extension of K.

                                          A field K is pseudo real closed if, for every absolutely irreducible variety V defined over K, if V has a Kr-rational point for every real closure Kr of K, then V has a K-rational point.
                                          We assume here that the pseudo real closed field K is not real closed or algebraically closed, which ensures the independence property, and also that K has at least one order, which ensures the strict order property (a pseudo real closed field with no orders is pseudo algebraically closed).

                                          See .
                                          strongly minimal
                                          o-minimal
                                          ω-stable, dp-minimal, and not strongly minimal
                                          superstable, dp-minimal, and not ω-stable
                                          stable, dp-minimal, and not superstable
                                          dp-minimal, SOP, and not o-minimal
                                          IP, SOP, NTP2
                                          supersimple and unstable
                                          simple and unstable, but not supersimple
                                          NTP2, TP1, and NSOP3
                                          unknown

                                          It is unknown whether or not the implication SOP3 ⇔ TP1 is strict.
                                          NTP2, SOP3, and NSOP4
                                          unknown

                                          It is unknown whether or not the strong order property hierarchy is strict inside NTP2.
                                          NTP2, SOPn, and NSOPn+1 for some
                                          n ≥ 4,
                                          unknown

                                          It is unknown whether or not the strong order property hierarchy is strict inside NTP2.
                                          NTP2 and NSOP, and SOPn for all n ≥ 1
                                          unknown

                                          It is unknown whether or not the strong order property hierarchy is strict inside NTP2.
                                          NTP2, NSOP, and SOP
                                          unknown

                                          It is unknown whether or not the strong order property hierarchy is strict inside NTP2.
                                          NSOP1 and TP2
                                          unknown

                                          It is unknown whether or not the implication SOP1 ⇔ TP is strict.
                                          NTP1 and SOP1
                                          unknown

                                          It is unknown whether or not the implication TP1 ⇔ SOP1 is strict.
                                          TP2, TP1, and NSOP3
                                          unknown

                                          It is unknown whether or not the implication SOP3 ⇔ TP1 is strict.
                                          TP2, SOP3, and NSOP4
                                          TP2, SOPn, and NSOPn+1 for some
                                          n ≥ 3
                                          To keep this section distinct from the others, we would technically want to say n ≥ 4. However every known example of a theory that is SOPn and NSOPn+1 for some n ≥ 4, has some analogue for any n ≥ 3.
                                          TP2, NSOP, and SOPn for all n ≥ 1
                                          TP2, SOP, and NSOP
                                          TP2 and SOP
                                          NIP, and not dp-minimal
                                          ω-stable, and not dp-minimal
                                          superstable, not ω-stable, and not dp-minimal
                                          stable, not superstable, and not dp-minimal