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)
Abstract
The 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 modeltheoretically, 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’ prototheory 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’ prototheory 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),
xy(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 ZFC2structures. We could also have kept going and
presented a sub-theory of it, viz., the theory of ZFCstructures. « 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.