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Much as group theory is the study of groups, or graph theory is the study of graphs, model theory is the study of models (also known as structures) of some language {{\mathcal L}} (which, in this post, will always be a single-sorted, first-order language). A structure is a set {X}, equipped with one or more operations, constants, and relations. This is of course an extremely general type of mathematical object, but (quite remarkably) one can still say a substantial number of interesting things about very broad classes of structures.

We will observe the common abuse of notation of using the set {X} as a metonym for the entire structure, much as we usually refer to a group {(G,1,\cdot,()^{-1})} simply as {G}, a vector space {(V, 0, +, \cdot)} simply as {V}, and so forth. Following another common bending of the rules, we also allow some operations on structures (such as the multiplicative inverse operation on a group or field) to only be partially defined, and we allow use of the usual simplifying conventions for mathematical formulas (e.g. writing {a+b+c} instead of {(a+b)+c} or {a+(b+c)}, in cases where associativity is known). We will also deviate slightly from the usual practice in logic by emphasising individual structures, rather than the theory of general classes of structures; for instance, we will talk about the theory of a single field such as {{\bf R}} or {{\bf C}}, rather than the theory of all fields of a certain type (e.g. real closed fields or algebraically closed fields).

Once one has a structure {X}, one can introduce the notion of a definable subset of {X}, or more generally of a Cartesian power {X^n} of {X}, defined as a set {E \subset X^n} of the form

\displaystyle  E = \{ (x_1,\ldots,x_n): P(x_1,\ldots,x_n) \hbox{ true} \} \ \ \ \ \ (1)

for some formula {P} in the language {{\mathcal L}} with {n} free variables and any number of constants from {X} (that is, {P(x_1,\ldots,x_n)} is a well-formed formula built up from a finite number of constants {c_1,\ldots,c_m} in {X}, the relations and operations on {X}, logical connectives such as {\neg}, {\wedge}, {\implies}, and the quantifiers {\forall, \exists}). Thus, for instance, in the theory of the arithmetic of the natural numbers {{\bf N} = ({\bf N}, 0, 1, +, \times)}, the set of primes {{\mathcal P}} is a definable set, since we have

\displaystyle  {\mathcal P} = \{ x \in {\bf N}: (\exists y: x=y+2) \wedge \neg (\exists z,w: x = (z+2)(w+2)) \}.

In the theory of the field of reals {{\bf R} = ({\bf R}, 0, 1, +, -, \times, ()^{-1})}, the unit circle {S^1} is an example of a definable set,

\displaystyle  S^1 = \{ (x,y) \in {\bf R}^2: x^2+y^2 = 1 \},

but so is the the complement of the circle,

\displaystyle  {\bf R}^2 \backslash S^1 = \{ (x,y) \in {\bf R}^2: \neg(x^2+y^2 = 1) \}

and the interval {[-1,1]}:

\displaystyle  [-1,1] = \{ x \in {\bf R}: \exists y: x^2+y^2 = 1\}.

Due to the unlimited use of constants, any finite subset of a power {X^n} of any structure {X} is, by our conventions, definable in that structure. (One can of course also consider definability without parameters (also known as {0}-definability), in which arbitrary constants are not permitted, but we will not do so here.)

We can isolate some special subclasses of definable sets:

  • An atomic definable set is a set of the form (1) in which {P()} is an atomic formula (i.e. it does not contain any logical connectives or quantifiers).
  • A quantifier-free definable set is a set of the form (1) in which {P()} is quantifier-free (i.e. it can contain logical connectives, but does not contain the quantifiers {\forall, \exists}).

Example 1 In the theory of a field such as {{\bf R}}, an atomic definable set is the same thing as an affine algebraic set (also known as an affine algebraic variety, with the understanding that varieties are not necessarily assumed to be irreducible), and a quantifier-free definable set is known as a constructible set; thus we see that algebraic geometry can be viewed in some sense as a special case of model theory. (Conversely, it can in fact be quite profitable to think of model theory as an abstraction of algebraic geometry; for instance, the concepts of Morley rank and Morley degree in model theory (discussed in this previous blog post) directly generalises the concepts of dimension and degree in algebraic geometry.) Over {{\bf R}}, the interval {[-1,1]} is a definable set, but not a quantifier-free definable set (and certainly not an atomic definable set); and similarly for the primes over {{\bf N}}.

A quantifier-free definable set in {X^n} is nothing more than a finite boolean combination of atomic definable sets; in other words, the class of quantifier-free definable sets over {X} is the smallest class that contains the atomic definable sets and is closed under boolean operations such as complementation and union (which generate all the other boolean operations). Similarly, the class of definable sets over {X} is the smallest class that contains the quantifier-free definable sets, and is also closed under the operation of projection {\pi_n: E \mapsto \pi_n(E)} from {X^{n+1}} to {X^n} for every natural number {n}, where {\pi_n: X^{n+1} \rightarrow X^n} is the map {\pi_n(x_1,\ldots,x_n,x_{n+1}) := (x_1,\ldots,x_n)}.

Some structures have the property of enjoying quantifier elimination, which means that every definable set is in fact a quantifier-free definable set, or equivalently that the projection of a quantifier-free definable set is again quantifier-free. For instance, an algebraically closed field {k} (with the field operations) has quantifier elimination (i.e. the projection of a constructible set is again constructible); this fact can be proven by the classical tool of resultants, and among other things can be used to give a proof of Hilbert’s nullstellensatz. (Note though that projection does not necessary preserve the property of being atomic; for instance, the projection of the atomic set {\{ (x,y) \in k^2: xy=1 \}} is the non-atomic, but still quantifier-free definable, set {\{ x \in k: \neg (k=0) \}}.) In the converse direction, it is not difficult to use the nullstellensatz to deduce quantifier elimination. For theory of the real field {{\bf R}}, which is not algebraically closed, one does not have quantifier elimination, as one can see from the example of the unit circle (which is a quantifier-free definable set) projecting down to the interval {[-1,1]} (which is definable, but not quantifer-free definable). However, if one adds the additional operation of order {<} to the reals, giving it the language of an ordered field rather than just a field, then quantifier elimination is recovered (the class of quantifier-free definable sets now enlarges to match the class of definable sets, which in this case is also the class of semi-algebraic sets); this is the famous Tarski-Seidenberg theorem.

On the other hand, many important structures do not have quantifier elimination; typically, the projection of a quantifier-free definable set is not, in general, quantifier-free definable. This failure of the projection property also shows up in many contexts outside of model theory; for instance, Lebesgue famously made the error of thinking that the projection of a Borel measurable set remained Borel measurable (it is merely an analytic set instead). Turing’s halting theorem can be viewed as an assertion that the projection of a decidable set (also known as a computable or recursive set) is not necessarily decidable (it is merely semi-decidable (or recursively enumerable) instead). The notorious P=NP problem can also be essentially viewed in this spirit; roughly speaking (and glossing over the placement of some quantifiers), it asks whether the projection of a polynomial-time decidable set is again polynomial-time decidable. And so forth. (See this blog post of Dick Lipton for further discussion of the subtleties of projections.)

Now we consider the status of quantifier elimination for the theory of a finite field {F}. If interpreted naively, quantifier elimination is trivial for a finite field {F}, since every subset of {F^n} is finite and thus quantifier-free definable. However, we can recover an interesting question in one of two (essentially equivalent) ways. One is to work in the asymptotic regime in which the field {F} is large, but the length of the formulae used to construct one’s definable sets stays bounded uniformly in the size of {F} (where we view any constant in {F} as contributing a unit amount to the length of a formula, no matter how large {F} is). A simple counting argument then shows that only a small number of subsets of {F^n} become definable in the asymptotic limit {|F| \rightarrow \infty}, since the number of definable sets clearly grows at most polynomially in {|F|} for any fixed bound on the formula length, while the number of all subsets of {|F|^n} grows exponentially in {n}.

Another way to proceed is to work not with a single finite field {F}, or even with a sequence {F_m} of finite fields, but with the ultraproduct {F = \prod_{m \rightarrow p} F_m} of a sequence of finite fields, and to study the properties of definable sets over this ultraproduct. (We will be using the notation of ultraproducts and nonstandard analysis from this previous blog post.) This approach is equivalent to the more finitary approach mentioned in the previous paragraph, at least if one does not care to track of the exact bounds on the length of the formulae involved. Indeed, thanks to Los’s theorem, a definable subset {E} of {F^n} is nothing more than the ultraproduct {E = \prod_{m \rightarrow p} E_m} of definable subsets {E_m} of {F_m^n} for all {m} sufficiently close to {p}, with the length of the formulae used to define {E_m} uniformly bounded in {m}. In the language of nonstandard analysis, one can view {F} as a nonstandard finite field.

The ultraproduct {F} of finite fields is an important example of a pseudo-finite field – a field that obeys all the sentences in the languages of fields that finite fields do, but is not necessarily itself a finite field. The model theory of pseudo-finite fields was first studied systematically by Ax (in the same paper where the Ax-Grothendieck theorem, discussed previously on this blog, was established), with important further contributions by Kiefe, by Fried-Sacerdote, by two papers of Chatzidakis-van den Dries-Macintyre, and many other authors.

As mentioned before, quantifier elimination trivially holds for finite fields. But for infinite pseudo-finite fields, such as the ultraproduct {F = \prod_{m \rightarrow p} F_m} of finite fields with {|F_m|} going to infinity, quantifier elimination fails. For instance, in a finite field {F_m}, the set {E_m := \{ x \in F_m: \exists y \in F_m: x=y^2 \}} of quadratic residues is a definable set, with a bounded formula length, and so in the ultraproduct {F =\prod_{m \rightarrow p} F_m}, the set {E := \prod_{m\rightarrow p} E_m} of nonstandard quadratic residues is also a definable set. However, in one dimension, we see from the factor theorem that the only atomic definable sets are either finite or the whole field {F}, and so the only constructible sets (i.e. the only quantifier-free definable sets) are either finite or cofinite in {F}. Since the quadratic residues have asymptotic density {1/2} in a large finite field, they cannot form a quantifier-free definable set, despite being definable.

Nevertheless, there is a very nice almost quantifier elimination result for these fields, in characteristic zero at least, which we phrase here as follows:

Theorem 1 (Almost quantifier elimination) Let {F} be a nonstandard finite field of characteristic zero, and let {E \subset F^n} be a definable set over {F}. Then {E} is the union of finitely many sets of the form

\displaystyle  E = \{ x \in V(F): \exists t \in F: P(x,t) = 0 \} \ \ \ \ \ (2)

where {V(F)} is an atomic definable subset of {F^n} (i.e. the {F}-points of an algebraic variety {V} defined over {F} in {F^n}) and {P: F^{n+1} \rightarrow F} is a polynomial.

Results of this type were first obtained essentially due to Catarina Kiefe, although the formulation here is closer to that of Chatzidakis-van den Dries-Macintyre.

Informally, this theorem says that while we cannot quite eliminate all quantifiers from a definable set over a nonstandard finite field, we can eliminate all but one existential quantifier. Note that negation has also been eliminated in this theorem; for instance, the definable set {F \backslash \{0\} = \{ x \in F: \neg(x=0) \}} uses a negation, but can also be described using a single existential quantifier as {\{ x \in F: \exists t: xt = 1 \}}.) I believe that there are more complicated analogues of this result in positive characteristic, but I have not studied this case in detail (Kiefe’s result does not assume characteristic zero, but her conclusion is slightly different from the one given here). In the one-dimensional case {n=1}, the only varieties {V} are the affine line and finite sets, and we can simplify the above statement, namely that any definable subset of {F} takes the form {\{ x\in F: \exists t \in F: P(x,t) = 0 \}} for some polynomial {P} (i.e. definable sets in {F} are nothing more than the projections of the {F}-points of a plane curve).

There is an equivalent formulation of this theorem for standard finite fields, namely that if {F} is a finite field and {E \subset F^n} is definable using a formula of length at most {M}, then {E} can be expressed in the form (2) with the degree of {P} bounded by some quantity {C_{M,n}} depending on {M} and {n}, assuming that the characteristic of {F} is sufficiently large depending on {M, n}.

The theorem gives quite a satisfactory description of definable sets in either standard or nonstandard finite fields (at least if one does not care about effective bounds in some of the constants, and if one is willing to exclude the small characteristic case); for instance, in conjunction with the Lang-Weil bound discussed in this recent blog post, it shows that any non-empty definable subset of a nonstandard finite field has a nonstandard cardinality of {(\alpha + O(|F|^{-1/2})) |F|^d} for some positive standard rational {\alpha} and integer {d}. Equivalently, any non-empty definable subset of {F^n} for some standard finite field {F} using a formula of length at most {M} has a standard cardinality of {(\alpha + O_{M,n}(|F|^{-1/2})) |F|^d} for some positive rational of height {O_{M,n}(1)} and some natural number {d} between {0} and {n}. (For instance, in the example of the quadratic residues given above, {d} is equal to {1} and {\alpha} equal to {1/2}.) There is a more precise statement to this effect, namely that the Poincaré series of a definable set is rational; see Kiefe’s paper for details.

Below the fold I give a proof of Theorem 1, which relies primarily on the Lang-Weil bound mentioned above.

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I recently discovered a CSS hack which automatically makes wordpress pages friendlier to print (stripping out the sidebar, header, footer, and comments), and installed it on this blog (in response to an emailed suggestion). There should be no visible changes unless one “print previews” the page.

In order to prevent this post from being totally devoid of mathematical content, I’ll mention that I recently came across the phenomenon of nonfirstorderisability in mathematical logic: there are perfectly meaningful and useful statements in mathematics which cannot be phrased within the confines of first order logic (combined with the language of set theory, or any other standard mathematical theory); one must use a more powerful language such as second order logic instead. This phenomenon is very well known among logicians, but I hadn’t learned about it until very recently, and had naively assumed that first order logic sufficed for “everyday” usage of mathematics.

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