Which model theory ?Ph. de Rouilhan

Presented on June 2, 2010; revised and expanded diapos.ppt, June 3, 2010

Conference Philosophy and model theory, June 2-5, 2010 (Université

Paris-Ouest and ENS)

AbstractThe term brings to mind the standard form that model theory began to take in the 1950s in the hands of Tarski and some other top flight logicians. The idea goes back to 19th century algebra of logic and the first major theorems of the early decades of the 20th century (starting in 1915). But, mathematicians did not need that tradition to think model-theoretically, as is clearly apparent in works on foundations like, for example, Hilbert's Grundlagen der Geometrie (1st ed.1900; 10th 1968), or, later, the Éléments de mathématique de Bourbaki (from 1939 to 1998). And the model theory that can be drawn from these works is no way identical with that of the logicians. It is enlightening to compare the two. This enables one to situate the logician's model theory exactly in the place due it de jure, if not de facto, in the architecture of mathematics and in the process to refute a good number of the idées reçues with regard to it.

§0. Introduction (1)

In a conference devoted to Philosophy and model theory, I should have entitled my talk: « Mathematics and model theory ». My question is of a philosophically important form: Where is what? Here: Where is model theory w.r.t. mathematics (and reciprocally)? Assuming that mathematics has been reduced to set theory, say ZFC or ZFC2, the kind of answer too often heard from logicians is: Mathematics is just one of the theories to which model theory (as a part of first- or second-order logic) can be straightforwardly applied.

§0. Introduction (2)

This answer raises difficulties, for the intended interpretation (in the non technical sense) of ZFC or ZFC2 is NOT a model of the axioms in the first place – not even a structure of interpretation (or model-structure) of the language!

§0. Introduction (3)

My aim is to give model theory its rationale and its right place INSIDE the architecture of mathematics.

§0. Introduction (4)

Bibliographic references: [1] N. Bourbaki, Eléments…, Livre 1, chap. 4

(1957)[2] Ph. de R., « La théorie des modèles et

l’architecture des mathématiques », in P. Gochet and Ph. de R., Logique épistémique and philosophie des mathématiques, Vuibert, 2007, pp. 39-118

§1. A proto-theory of models (BT) (1)

BT (« B » like « Bolzano », and « T » like « Tarski) is outlined in Tarski, « On the concept of logical consequence », 1936; it is reminiscent of Bolzano’s Wissenschaftslehre, 1937, that Tarski didn’t know. It takes place in the frame of a logica magna, say ZFC or ZFC2, and its object-languages take place in the frame of ZFC. NB I am using the phrase « In the frame of a (language of a) theory » in the sense of « In the language obtained from the language of the theory by adding extralogical constants ».

§1. A proto-theory of models (BT) (2)

• As a typical object-language for us, here, let M (sic) be an extension of the language of ZFC obtained by adding to it a finite sequence of objectual constants (singular terms), a1, …, an.

• An interpretation, I, of M is a sequence of n objects, x1, …, xn.

• A statement A of M is true in I iff it becomes true when a1, …, an are assigned references x1, …, xn, respectively. N.B. I skip the definition of truth simpliciter and ignore the difficulties related to the Liar paradox. In order to overcome these difficultie, one can either go up to an logically richer frame, like ZFC2, and explicitely define truth for M, or to withdraw to a recursive or a schematic definition in

the frame of ZFC, with all that such a move implies.

§1. A proto-theory of models (BT) (3)

• A model of a set, E, of statements of M is an interpretation in which the statements of E are true.

• A is logical consequence of E iff every model of E is a model of A.

• Etc.

§2. Mathematics is theories of species of structure

Mathematical theories can be described as theories of species of structure. The §3 is devoted to structures, the §4, to species of structure,the § 5 to theories of species of structure.

§3. Structures (1)

Example 1. Let G, e, be s.t. 1°) set(G), 2°) e G,

: G G G.Then, G ; e, is a structure; G is its basic set, and e and its structural components.

§3. Structures (2)

Example 2. Let N, 0, S be s.t. 1°) Set(N), 2°) 0 N,

S : N N. Then, N ; 0, S is a structure; N is its basic set, and 0 and S its structural components.

§3. Structures (3)

Generalization. Let U1, …, Un,V1, …, Vp s.t. 1°) U1, …, Un are sets, 2°) V1, …, Vp are members of certain (finite) types in the hierarchy of sets based on U1, …, Un. Then U1, …, Un ; V1, …, Vp is a structure; U1, …, Un are its basic sets, and V1, …, Vp its structural components.N.B. Fore the sake of simplicity, here, I define the concept of structure only in a restricted a sense, which will lead to a concept of species of structure not allowing, for example, for the species of vector spaces on a fixed field. For an adequate generalization, see the bibliography given above.

§4. Species of structure (1)

Example 1. The class of all structures G ; e, s.t. 1°) set(G), 2°) e G and : G G G, (as in the first example of structure given above, §3, n°1), and 3°) (x, y, z G)((x y) z = x (y z)),

(x G)(x e = e x = x),(x G)(y G)(x y = y x = e)is a species of structure (viz. groups).

§4. Species of structure (2)

Example 2. The class of all structures N ; 0, S 1°) set(N), 2°) 0 N and S : S S (as in the second

example of structure above, §3, n°2), and 3°) (x, y N)(x ≠ y S(x) ≠ S(y)),(x N)(0 ≠ S(x)),(y N)((0 y & (x N)(x y S(x) y)) N y),is a species of structure (viz. simply infinite sets, or progressions).

§4. Species of structure (3)

Remark. In examples 1 and 2, axioms under 3° express properties of structures under consideration which are structural, formal, independant of the nature of the elements of the basic sets. As Bourbaki puts it, these axioms are transportable (w.r.t. the kind of structure determined by axioms under 1° and 2°) That notion could be rigourously defined in terms of invariance of a property under isomorphism (w.r.t. the kind of structure determined by axioms under 1° and 2°)

§4. Species of structure (4)

Generalization. The class of all structures U1, …, Un ; V1, …, Vp, with n and p fixed, s.t. 1°) U1, …, Un are sets,

2°) V1, …, Vp are members of certain fixed (finite) types in the hierarchy of sets based on U1, …, Un, as in the general definition of a structure above (§3, n°3), and 3°) [here finitely many transportable axioms (transportable: w.r.t. the kind of structure determined by axioms under 1° and 2°)]is a species of structure.

§5. Theories of species of structure (1)

Example 1. The theory obtained from ZFC by adding objectual constants (singular terms) « G », « e », «  » and axioms under 1°, 2°, and 3° given above in the first example of species of structure (§4, n°1), is the theory of the species of structure of group (alias group theory). « G » : basic constant; « e », «  » : structural constants; axiom under 1°: axiom of position;axioms under 2°: axioms of typification;axioms under 3°: axioms of specification.

§5. Theories of species of structure (2)

Example 2. The theory obtained from ZFC by adding objectual constants (singular terms) « N », « 0 », « S » and axioms under 1°, 2°, and 3° given above in the second example of species of structure (§4, n°2), is the theory of the species of structure of progressions (alias progression theory). « N » : basic constant; « 0 », « S » : structural constants; axiom under 1°: axiom of position;axioms under 2°: axioms of typification;axioms under 3°: axioms of specification.

§5. Theories of species of structure (3)

Generalization. The theory obtained from ZFC by adding an ordered pair of sequences of objectual constants a1, …, an ; b1, …, bp (with n, p explicitly given), and of axioms like to those given above under 1°, 2°, and 3° in the general definition of a species of structure (§4, n°4) is a theory of species of structure. a1, …, an : basic constants; b1, …, bp: structural constants; axioms under 1°: axioms of position;axioms under 2°: axioms of typification; axioms under 3°: axioms of specification).

§6. Mathematicians’ proto-theory of models (MT1) (1)

MT1 is a variant of BT: it is obtained from BT mutatis mutandis:a1, …, an is to be replaced by

a1, …, an ; b1, …, bp;

x1, …, xn is to be replaced by

x1, …, xn ; y1, …, yp;

etc.

§6. Mathematicians’ proto-theory of models (MT1) (2)

Example. In terms of MT1, a group is a model of the proper axioms (those of position, of typification and of specification) of group theory. The common properties of groups are the logical consequences of the proper axioms of the theory. The specialist of groups limits himself to those which are transportable. He may also be interested in questions of relative consistency or independance, or of categoricity, etc., (in the MT1 sense). Etc.

§7. The origin of logicians’ (inclusive version of) model theory (MT2) (1)Seminal remark for MT2. The axioms of specification of usual theories of species of structure can be formulated in a langage, L, much simpler than M. In the general case, L will be a n-sorted, typed, language. The basic constants and the axioms of position and of typification as such will disappear, their content being built in the new notion of interpretation for L. The notion of transportability will become idle, for every statement of L will be transportable (in the relevant sense).

§7. The origin of logicians’ (inclusive version of) model theory (MT2) (2)Example 1. For group theory, the axioms are now formulated thus:(x, y, z)((x y) z = x (y z)),

x(x e = e x = x),xy(x y = y x = e),with variables ranging over G; « e », singular term w.r.t. G; « », dyadic functor w.r.t. G. No basic constant, no axiom of position, no axioms of typification: they have disappeared, their content is built in the new notion of interpretation for L.

§7. The origin of logicians’ (inclusive version of) model theory (MT2) (3)Example 2. For progression theory, the axioms are now formulated thus:(x, y)(x ≠ y Sx ≠ Sy),x(0 ≠ Sx),X((X0 & x(Xx XSx)) xXx),with lower case variables ranging over N, and upper case ones over P(N); « 0 », singular term w.r.t. N; « S », monadic functor w.r.t. N.No basic constant, no axiom of position, no axioms of typification: they have disappeared, their content is built in the new notion of interpretation for L.

§7. The origin of logicians’ (inclusive version of) model theory (MT2) (4)Generalization. There are no basic constants, no axioms of position or of typification, no notion of transportability. The axioms of specification are now formulated in a classical, n-sorted, - order L, with a sequence of p constants, 1, …,p, of types corresponding to the old « types » of b1, …,bp. A variable of the sort i will range over the generic sets designated by ai (1 ≤ i ≤ n). It’s easy to define the effective translation function, * : L M, applying the new axioms on the old ones.

§8. Comparison of MT1 and MT2 (1)

Pointing out that in MT1 was a « fixed domain », while MT2 is a « variable domain », model theory would be superficial and misleading. Idem for the possible remark that MT2 is a model theory with, and MT2 was a model theory without, « cross restrictions » (Etchemendy). MT1 and MT2 must be compared when applied to the study of the same structures, and not otherwise. Then, the following theorem settles the question.

§8. Comparison of MT1 and MT2 (2)

Theorem of cooperation. – Let’s note Pos and Typ the axioms of position and of typification, respectively, in M; and ⊨MT1 and

⊨MT2, the relations of logical consequence in the sense of MT1 and MT2 respectively. Then, for any set, E {A}, of statements of L,

E ⊨MT2 A iff Pos Typ E* ⊨MT1 A*.

§9. Back to the beginning (1)

In §5, we could have added a third example of theory of species of structure: a theory with axioms of specification analogous to those of ZF2, but with, mandatorily, a distinctive notation (e.g., « SET », not to be confused with « set », « BELONG » not to be confused with «  »). Call it the MT1-version of theory of ZFC2-structures.

§9. Back to the beginning (2)Then, in §7, we could have presented the corresponding MT2-version of the theory of ZFC2-structures. We could also have kept going and presented a sub-theory of it, viz., the theory of ZFC-structures. « set » and « » would not have occured in the langage of these theories. So, we could have go back to using the usual set-theoretical notation, by replacing « SET » and « BELONG » by « set » and « », respectively. So, the variables being litterally the same, the axioms of specification of our theories of ZFC- and of ZFC2-structures would have become litterally indiscernible from axioms of ZFC and of ZFC2 themselves.

§9. Back to the beginning (3)

Diagnosis. The idea that model theory (for us MT2) is straightforwardly applicable to ZFC and ZFC2 stems from a confusion between –or, at least, the bad habit of speaking ambiguously of– these theories themselves and the counterparts of them that we have just contemplated.