Functorial Models, Horn

Products, and Positive Omitting Types Realizability

Cyrus F NouraniAkdmkrd.tripod;com

http://www.lama.univ-savoie.fr/~hyvernat/Realisabilite2012/index.php June 5, 2012

Copyright © 2006 Photo reproduction for noncommercial use and confernce or journal publications is permitted

without payment of royalty provided the reference and copyright notice are included on the first page. *Affiliations Academia California http://beam.to/af2im Academic Email:

Cyrus F.NouraniAkdmkrd.tripod.com

Abstract Infinitary positive language categories are defined and infintary complements to Robinson consistency from the author’s preceding papers are further developed to present new positive omitting types techniques. The author (1994) defined a preorder small-complete category on the Kiesler fragments. Based on the infinitiary counterpart Robinson’s consistency functorial Fragment Limit chain models, were defined (1996). Positive local realizability allows us to present new omitting type theorems, positive morphic extensions, product Horn models and new consistency techniques, positive categories and Horn categories that are are new fragment categories (Nourani 2005).

AbstractInfinitary positive language categories are defined and infintary complements to Robinson consistency from the author’s preceding papers are further developed to present new positive omitting types techniques. The author (1994) defined a preorder small-complete category on the Kiesler fragments. Based on the infinitiary counterpart Robinson’s consistency functorial Fragment Limit Chain models were defined (1996). Positive local realizability allows us to present new omitting type theorems, positive morphic extensions, endextension, product Horn models and new consistency techniques, positive categories and Horn categories that are arenew fragment categories (Nourani 2005).

The PreliminariesThe Preliminaries

Functorial Model Theory and Infinitary Language Categories, Proc. ASL, Functorial Model Theory and Infinitary Language Categories, Proc. ASL,

January 1995, San Francisco. January 1995, San Francisco.

See BSL, Vol.2,Number 4, December 1996.See BSL, Vol.2,Number 4, December 1996.

Functorial Model Theory and Generic Fragment Consistency Models,Functorial Model Theory and Generic Fragment Consistency Models,

October 1996, AMS-ASL, San Diego, January 1997.October 1996, AMS-ASL, San Diego, January 1997.www.math.ucla.edu/~asl/bsl/0303/0303-006.ps

Nourani,C.F., "Functorial Consistency, May 1997, AMS Nourani,C.F., "Functorial Consistency, May 1997, AMS

927,Millwaukee,Wisconsin,Abstract number 927-03-29927,Millwaukee,Wisconsin,Abstract number 927-03-29 Functorial Metamathematics, Maltsev Meeting, Novosibirsk, Russia, Functorial Metamathematics, Maltsev Meeting, Novosibirsk, Russia,

November 1998November 1998

The Keisler FragmentThe Keisler Fragment

Start with a well-behaved countable fragment of an infinitary language L1, defined by H.J. Keisler. The term fragment is not inconsistent with the terminology applied in categorical logic. A subclass F of class of all formulas of an Infinitary language is called a fragment, if (a) for each formula in F all the subformulas of also belong to F; and (b) F is closed under substitution: if is in F, t is a term of L, x is a free variable in , the (x/t) is in F. Definition By a fragment of L1, we mean a set L<A> of formulas such that (1) Every formula of L belongs to L<A> (2) L<A> is closed under , x, and finite disjunction (3) if (x) L<A> and t is a term then (t) L<A> (4) If L<A> then every subformula of L<A> Subformulas are defined by recursion from the infinite disjunction by sub (V) = sub () {V taken over the set of formulas in ; with the basis defined for atomic formulas} by sub() = {}; and for compound formulas by taking the union of the subformulas quantified or logically connected with the subformula relation applied to the original formula as a singleton. L is a countable language for first order logic. Let’s call the fragment L1,K.

Infinitary Language CategoriesInfinitary Language Categories

We take the Op category formed by L1,B. Present a functor to take the Syntax Logische der Sprache from this category to the category Set, creating a limit corresponding to a model for the language defined from the syntax.te category. [] Op A functor F: L1,B Set is defined by a list of sets Mn and functions fn. We refer to F by the generic model functor since it defines a generic model from language strings. The functor F is a list of sets Mn corresponding to a free structure on L1,B. The functions fn : Fn+1 Fn. A generic model is defined by the limit on F.

IFLCs are not the same as a categorical interpretation for logic in the categorical logic sense. Thus IFLCs have their own theoretical properties and application areas defined by the present author since July 1994 <ECCT>. We define a model theory for functors which are really what Logische Syntax der Sprache on an Infinitary language L1, might be. Starting with the well-behaved countable fragment of an infinitary language L1, , we called L1,B.

IFLCSIFLCS

Generic FunctorsGeneric Functors

Op Define a functor F: L1,B Set by a list of sets Mn and functions fn. Let us refer to the above functor by the generic model functor since it defines generic sets from language strings to form limits and models. The model theoretic properties are not defined in the present paper and are to be presented elsewhere by this author. The proof for the following theorem follows from the sort of techniques applied by the present author to define generic sets with the fragment L1,B

by this author 1982, 1983. Theorem The generic model functor has a limit. Theorem The Generic model functor defines a positive generic model.

Generic Functors and IFLCSGeneric Functors and IFLCSCreating limits amounts to defining generic sets on L1, for L1,K. Since the present theory creates limits for Infinitary language sets to define models. The techniques have been applied by this author to define Positive Forcing [1980-81]. Let us refer to the above functor by the generic model functor since it defines generic sets from language strings to form limits and thus models. The following theorem presented itself in the course

oof writing the proofs for the present paper and is sstated here is a consequence of the postive forcing results proved by this author in [1980's]. Theorem The power set of an inductive theory defined on L1,B generates a positive generic model for the theory, provided the inductive theory has a defined generic diagram for its initial model.

String Models and LimitsString Models and Limits

The functor F is a list of sets Fn, consisting of

(a) the sets corresponding to an initial structure on L1,K, for example, to f(t1,t2,.. tn) in L1,K there corresponds the equality relation f(t1,...,tn)=ft1...tn in Set;

(b) the functions fi:Fi+1 Fi. Form the product set iFi. Let l = {l0,l1,l2,....}, with each ln Fn, and its projections pn: iFi Fn, forming the following diagram. F0 F1 Fn Fn+1 iFi include M = Lim F To form a limit at a vertex for commuting with the projections take the subset S of the strings in S, which match under f , i.e. f sn+1 = sn, for all n.

Robinson ConsistencyRobinson Consistency Theorem Theorem (Robinson’s Consistency Theorem) Let (Robinson’s Consistency Theorem) Let L1, L2 L1, L2 be be

two languages. Let two languages. Let L = L1 L = L1 L2. L2. Suppose T is a complete Suppose T is a complete theory in theory in L L and T1 and T1 T, T2 T, T2 T are consistent in T are consistent in L1, L2, L1, L2, respectively. Then T1 respectively. Then T1 T2 is consistent in the language T2 is consistent in the language L1 L1 L2. L2.

We want to define functorial models piecemeal from We want to define functorial models piecemeal from language fragments by an infinite limit. There are two ways language fragments by an infinite limit. There are two ways to view it . to view it .

A- Take LA- Take L1, B1, B language fragments, define language fragments, define -chain models, -chain models, and back and forth to a limit diagram model.and back and forth to a limit diagram model.

B- Define models for the Fi from elementary diagrams, i.eB- Define models for the Fi from elementary diagrams, i.e define a limit model by embedding from a D<A,G> model.define a limit model by embedding from a D<A,G> model.

What complete theory can we fall onto? It has to be the What complete theory can we fall onto? It has to be the theory theory

Th(F), where F is the generic model functor, i.e. Th(F), where F is the generic model functor, i.e. OpOp Th (F: Th (F: L L 1, B1, B Set Set). ).

Elementary ChainsElementary Chains

An elementary chain is a chain of models A0 A1 A3 A4 ...A such that A < A whenever < < , < is the elementary extension relation. Elementary chains are applied to define Robinson’s consistency theorem and we plan to define similar techniques for defining String Models on the infinitary L1,K fragments. Of course the task is quite difficult and intricate functorial models on language fragments.

Fragment Consistency Fragment Consistency The following is our infinitary complement to Robinson The following is our infinitary complement to Robinson

ConsistencyConsistency

Theorem Fragment Model Chain ConsistencyTheorem Fragment Model Chain Consistency

Let T<i,i+1> be the complete theory in L<i,i+1> = Li intersect Let T<i,i+1> be the complete theory in L<i,i+1> = Li intersect Li+1, defined by Th (A(Fi) intersect A(Fi+1)). Let Ti and Ti+1 be Li+1, defined by Th (A(Fi) intersect A(Fi+1)). Let Ti and Ti+1 be arbitrary consistent theories for the sub fragments Li and Li+1, arbitrary consistent theories for the sub fragments Li and Li+1, respectively, satisfying Ti contains T<i,i+1> and Ti+1 contains respectively, satisfying Ti contains T<i,i+1> and Ti+1 contains T<i,i+1>. Let Ai be A(Fi) and Bi be A(Fi+1). Starting with a basis T<i,i+1>. Let Ai be A(Fi) and Bi be A(Fi+1). Starting with a basis model A0, A0 | L<0,1> and B0 | L<0,1> are models of a complete model A0, A0 | L<0,1> and B0 | L<0,1> are models of a complete theory, hence, A0 | L<0,1> theory, hence, A0 | L<0,1> B0 | <0,1>. It follows that the B0 | <0,1>. It follows that the elementary diagram of A0|l<0,1> is consistent with the elementary diagram of A0|l<0,1> is consistent with the elementary diagram of B0 | L<0,1>. elementary diagram of B0 | L<0,1>.

(i) There are iterated elementary extensions Bi+1> Bi and an (i) There are iterated elementary extensions Bi+1> Bi and an embedding fi: Ai < Bi embedding fi: Ai < Bi

(ii) A slalom between the language pairs Li Li+1 gates to a limit (ii) A slalom between the language pairs Li Li+1 gates to a limit model for L obtains a model M for Lmodel for L obtains a model M for L1,K 1,K ..

Fragment Consistency ModelsFragment Consistency Models OP A functor F: L1,K Set can be defined by sets Fi, where the Fi’s are defining a free structure on some subfragment of L1,B. To be specific we can define the subfragment models A(Fi) straight from the -inductive definition of the Infinitary fragment. F0 assigns names to the Set members, for example. F1 can define 1-place functions and relations, so on and so forth. Taking the languages defined before with the Robinson’s consistency theorem define Functorial Limit Chain models as follows. We shall refer to it by FLC-models.

FLC ModelsFLC Models

Let A and B, be models for Fi and Let A and B, be models for Fi and Fi+1,respectively. Let A Fi+1,respectively. Let A L B, and f: A <L B L B, and f: A <L B mean the L-reduct of A and B are mean the L-reduct of A and B are elementarily equivalent and that f is an elementarily equivalent and that f is an elementary embedding of A|L into B|L.elementary embedding of A|L into B|L.

Let A Let A FLCFLC model be the limit model defined model be the limit model defined by the elementary chain on the L-reducts of by the elementary chain on the L-reducts of the models defined by the Fi’s. A specific the models defined by the Fi’s. A specific FLC FLC model is defined by theorem ’s proof. model is defined by theorem ’s proof.

Theorem Theorem There is an elementary chain There is an elementary chain FLC FLC model for L model for L1,K. .

Fragment Model TowersFragment Model Towers

Let T<i,i+1> be the complete theory in L<i,i+1> = Li Li+1, defined by Th (A(Fi) A(Fi+1)). Let Ti and Ti+1 be arbitrary consistent theories for the subfragments Li and Li+1, respectively, satisfying Ti T<i,i+1> and Ti+1 T<i,i+1>. Let Ai be A(Fi) and Bi be A(Fi+1). Starting with a basis model A0, A0|L<0,1> and B0|L<0,1> are models of a complete theory, hence, A0|L<0,1> B0| l<0,1>. It follows that the elementary diagram of A0|l<0,1> is consistent with the elementary diagram of B0|L<0,1>. Therefore, there are elementary extensions B1> B0 and an embedding f1: A0 < B1 at L<01>. Passing to the expanded language L<0,1>A0, we have (A0,a)a A0 L<0,1> A0 (B1,fa) a A0. g1 inverse is an extension of f1. Iterating, we obtain the tower depicted sideways A0 < A1< A2<......... \ f1 |g1 \f2 |g2 B0 < B1 < b2<.....

The Alpine Language The Alpine Language SlalomSlalom

Slalom between the language pairs Li Li+1 gates to a limit model for L. For each m, fm inverse(gm) fm+1, fm: Am - 1 < Bm at L<m - 1,m>. Let A = Am, m < , B= Bm, m <. B is isomorphic to a model B' such that A | L = B' | L. Piecing A and B' together we obtain a model M for L1,B. Note that the language L varies in our proof, while it does not in the -chain model proof for Robinson’s consistency theorem. Thus the techniques are perhaps new.

A concluding theorem.

Theorem There are functorial elementary chain models.

FiltersFilters Let I be a nonempty set. Let S(I) be the set of all subsets of Let I be a nonempty set. Let S(I) be the set of all subsets of

I. A filter D over I is defined to be a set D < S(I) such that I. A filter D over I is defined to be a set D < S(I) such that I I D; if X,Y D; if X,Y D, hen X ∩ Y D, hen X ∩ Y D,; if X D,; if X D and X < Z < I, then D and X < Z < I, then Z Z D. D.Note that every filter D is a nonempty set since I Note that every filter D is a nonempty set since I D. D. example filters are the trivial filter D = {I}. The improper example filters are the trivial filter D = {I}. The improper filter D = S(I). filter D = S(I). For each Y < I, the filter D = {X <I; Y < X}; this filter is For each Y < I, the filter D = {X <I; Y < X}; this filter is called the principal filter generated by Y. D is said to be a called the principal filter generated by Y. D is said to be a proper filter iff it is not the improper filter S(I).proper filter iff it is not the improper filter S(I).

Let E be a subset of S(I). By the filter generated by E we Let E be a subset of S(I). By the filter generated by E we mean the intersection D of all filters over I which include E: mean the intersection D of all filters over I which include E: D = ∩ {F: E < F and F is a filer over I}. E is said to have the D = ∩ {F: E < F and F is a filer over I}. E is said to have the finite intersection property iff the intersection of any finite finite intersection property iff the intersection of any finite number if elements of E is nonempty.number if elements of E is nonempty.

Can prove that the filter D generated by E, any subset E of Can prove that the filter D generated by E, any subset E of S(I), is a filter over I.S(I), is a filter over I.

Fragment Horn ModelsFragment Horn Models Positive forcing had defined T* to be T Positive forcing had defined T* to be T

augmented with induction schemas on the augmented with induction schemas on the generic diagram functions.generic diagram functions.

Proposition Proposition Let I be the set T*. Let φ Let I be the set T*. Let φ (x1…xn) be a Horn formula and let (x1…xn) be a Horn formula and let i i i i I I be models for language L. let a1….an be models for language L. let a1….an i i I Ai. The I Ai. The i are fragment Horn models. i are fragment Horn models.

If {i If {i I: I: i ╞ φ[(a1(i) …an(i)]} then the i ╞ φ[(a1(i) …an(i)]} then the direct direct D D on on i ╞ φ[a1D …anD], where D is i ╞ φ[a1D …anD], where D is the generic filter on T*.the generic filter on T*.

Positive Morphisms and Horn ModelsPositive Morphisms and Horn Models

Definition Definition A formula is said to be positive iff it is built from A formula is said to be positive iff it is built from atomic formulas using only the connectives &, v and the atomic formulas using only the connectives &, v and the quantifiers quantifiers , , ..

Definition Definition A formula A formula (x1,x2,...,xn) is preserved under (x1,x2,...,xn) is preserved under homomorphisms iff for any homomorphisms f of a model A onto homomorphisms iff for any homomorphisms f of a model A onto a model B and all a1,...,an in A if A |= [a1,...,an] B ╞ a model B and all a1,...,an in A if A |= [a1,...,an] B ╞ [fa1,...,fan]. [fa1,...,fan].

Definition Definition A generic diagram for a structure M is a diagram A generic diagram for a structure M is a diagram D<A,G>, such that there is a proper diagram defined with a D<A,G>, such that there is a proper diagram defined with a specific function set. specific function set.

The function set might be The function set might be 1 Skolem functions for the set theory 1 Skolem functions for the set theory example.example.

Positive forcing had defined T* to be T augmented with Positive forcing had defined T* to be T augmented with induction schemas on the generic diagram functions.induction schemas on the generic diagram functions.

Proposition Proposition Let I be the set T*. Let φ (x1…xn) be a Horn Let I be the set T*. Let φ (x1…xn) be a Horn formula and let formula and let i i i i I be models for language L. let a1….an I be models for language L. let a1….an i i I Ai. The I Ai. The i are fragment Horn models. i are fragment Horn models.

If {i If {i I: I: i |= φ[(a1(i) …an(i)]} then the direct i |= φ[(a1(i) …an(i)]} then the direct D on D on i ╞ i ╞ φ[a1D …anD], where D is the generic filter on T*.φ[a1D …anD], where D is the generic filter on T*.

Fragment Consistent Models over Fragment Consistent Models over AlgebrasAlgebras

Applying the above definitions for positive formulas Applying the above definitions for positive formulas we can prove the following as a lemma and a we can prove the following as a lemma and a consequent theorem.consequent theorem.

Theorem Theorem The embedding to form elementary The embedding to form elementary chains on Fi’s can be defined by a back and forth chains on Fi’s can be defined by a back and forth model design from the strings in model design from the strings in L L 1,K 1,K language language fragments.fragments.

Theorem Theorem By defining models corresponding to the By defining models corresponding to the

Fi on the fragments as an Fi on the fragments as an -chain from the -chain from the elementary diagrams on the Th (A(Fi)) a generic elementary diagrams on the Th (A(Fi)) a generic model is defined by the limit. model is defined by the limit.

Consistency on the PositiveConsistency on the Positive Let us start from certain model-theoretic premises with the Let us start from certain model-theoretic premises with the

following two propositions known form basic model theory.following two propositions known form basic model theory.

Proposition Proposition Let Let and and BB be models for L. Then be models for L. Then is is isomorphically embedded in isomorphically embedded in BB iff iff BB can be expanded to a can be expanded to a model of the diagram of model of the diagram of ..

Proposition Proposition Let Let and and BB be models for L. Then be models for L. Then is is homomrphically embedded in homomrphically embedded in BB iff iff BB can be expanded to a can be expanded to a model of the positive diagram of model of the positive diagram of .. Let Σ be a set of Let Σ be a set of formulas in the variables x1…xn. Let formulas in the variables x1…xn. Let be a model for L. We be a model for L. We say that say that realizes Σ iff some n-tuple of elements of A realizes Σ iff some n-tuple of elements of A satisfies Σ in satisfies Σ in .. omits Σ iff omits Σ iff does not realize Σ. From does not realize Σ. From basic model theory we know that basic model theory we know that

Let Σ (x1…xn) be a set of formulas of L. A theory T in L is Let Σ (x1…xn) be a set of formulas of L. A theory T in L is said to locally realize Σ iff there is a formula φ (x1…xn) in L said to locally realize Σ iff there is a formula φ (x1…xn) in L s.t.s.t.

(i) φ is consistent with T, and (i) φ is consistent with T, and (ii) for all σ (ii) for all σ Σ, T ╞ φ Σ, T ╞ φ σ σThat is every n-tuple of T which realizes φ satisfies Σ.We say That is every n-tuple of T which realizes φ satisfies Σ.We say that T locally omits Σ iff T does not locally realize Σ.For our that T locally omits Σ iff T does not locally realize Σ.For our purposes we define a new realizability basis.purposes we define a new realizability basis.

Positive ConsistencyPositive Consistency Definition Definition Let Σ (x1…xn) be a set of formulas of L. Say that Let Σ (x1…xn) be a set of formulas of L. Say that

a positive theory T in L a positive theory T in L positively locally realizepositively locally realize Σ iff there is Σ iff there is a formula φ (x1…xn) in L s.t. (i) φ is consistent with T, anda formula φ (x1…xn) in L s.t. (i) φ is consistent with T, and

(ii) for all σ (ii) for all σ Σ, T ╞ φ or T Σ, T ╞ φ or T σ is not consistent. σ is not consistent.

Definition Definition Given models A and B, with generic diagrams DGiven models A and B, with generic diagrams DAA and Dand DB B we say that DA homomrphially extends D we say that DA homomrphially extends DBB iff there iff there is a homomorhic embedding f: A is a homomorhic embedding f: A B. B.

Theorem Theorem Let Let L1, L2 L1, L2 be two positive languages. Let be two positive languages. Let L = L1 L = L1 L2. L2. Suppose T is a complete theory in Suppose T is a complete theory in L L and T1 and T1 T, T2 T, T2 T are consistent in T are consistent in L1, L2, L1, L2, respectively. Suppose there is respectively. Suppose there is model M definable from a positive diagram in the language model M definable from a positive diagram in the language L1 L1 L2 such that L2 such that there are models M1 and M2 for T1 and there are models M1 and M2 for T1 and T2 where M can be homorphically embedded in M1 and M2. T2 where M can be homorphically embedded in M1 and M2.

(i) T1 (i) T1 T2 is consistent. T2 is consistent. (ii) There is model N for T1 (ii) There is model N for T1 T2 T2 definable from a positive definable from a positive

diagram that homomorphically extends that of M1 and M2diagram that homomorphically extends that of M1 and M2..

Positive Fragment CategoriesPositive Fragment Categories Let LLet LP,ωP,ω be the positive fragment obtained from be the positive fragment obtained from

the Kiesler fragment. the Kiesler fragment. Define the category LDefine the category LP,ωP,ω to be the category with to be the category with

objects positive fragments and arrows the sub objects positive fragments and arrows the sub formula preorder on formulas. formula preorder on formulas.

OP OP Define a functor F: L Define a functor F: LP,ω P,ω Set by a list of sets Mn and functions fn. The Set by a list of sets Mn and functions fn. The functor F is a list of sets Fn, consisting of functor F is a list of sets Fn, consisting of

(a) the sets corresponding to an initial structure (a) the sets corresponding to an initial structure on Lon LP,ωP,ω, for example the free syntax tree , for example the free syntax tree structure, where to f(t1,t2,.. tn) in Lstructure, where to f(t1,t2,.. tn) in LP,ωP,ω there there corresponds the equality relation corresponds the equality relation f(t1,...,tn)=ft1...tn in Set;f(t1,...,tn)=ft1...tn in Set;

(b) the functions fi:Fi+1 (b) the functions fi:Fi+1 Fi. Fi.

Positive Morphs and Fragment Positive Morphs and Fragment CategoriesCategories

Let LLet LP,ωP,ω be the positive fragment obtained from the Kiesler be the positive fragment obtained from the Kiesler fragment. fragment.

Define the category Define the category LLP,ωP,ω to be the category with objects positive to be the category with objects positive fragments and arrows the sub formula preorder on formulas. fragments and arrows the sub formula preorder on formulas.

OpOp Define a functor F: Define a functor F: L LP,ω P,ω - - Set Set by a list of sets Mn and by a list of sets Mn and

functions fn. The functor F is a list of sets Fn, consisting of functions fn. The functor F is a list of sets Fn, consisting of (a) the sets corresponding to an initial structure on L(a) the sets corresponding to an initial structure on LP,ωP,ω , for , for

example the free syntax tree structure, where to f(t1,t2,.. tn) example the free syntax tree structure, where to f(t1,t2,.. tn) in Lin LP,ωP,ω there corresponds the equality relation f(t1,...,tn)=ft1...tn there corresponds the equality relation f(t1,...,tn)=ft1...tn

in Set;in Set; (b) the functions fi:Fi+1 (b) the functions fi:Fi+1 Fi. Fi.

Proposition 4.9Proposition 4.9 (positive morphic extensions) The following (positive morphic extensions) The following are equivalent:are equivalent:

Every positive sentence holding on Every positive sentence holding on also holds on also holds on BB.. There are elementary extensions There are elementary extensions < < ’ , ’ , BB < < BB’ such that ’ such that BB’ ’

is a homomorphic image of is a homomorphic image of ’’

Infinitary Consistency Over AlgebrasInfinitary Consistency Over Algebras

TheoremTheorem Infinitary Fragment Consistency on Infinitary Fragment Consistency on Algebras Let T<i,i+1> be complete theories for Algebras Let T<i,i+1> be complete theories for L<i,i+1> = Li intersect Li+1. Let Ti and Ti+1 be L<i,i+1> = Li intersect Li+1. Let Ti and Ti+1 be arbitrary consistent positive theories for the sub arbitrary consistent positive theories for the sub fragments Li and Li+1, respectively, satisfying Ti fragments Li and Li+1, respectively, satisfying Ti contains T<i,i+1> and Ti+1 contains T<i,i+1>. contains T<i,i+1> and Ti+1 contains T<i,i+1>. Let Ai be A(Fi) and Bi be A(Fi+1). Let Ai be A(Fi) and Bi be A(Fi+1).

(i) There are iterated elementary extensions (i) There are iterated elementary extensions BBi+1> i+1> BBi and an embedding fi: Ai < Bi i and an embedding fi: Ai < Bi

(ii) Slalom between the language pairs Li Li+1 (ii) Slalom between the language pairs Li Li+1 gates to a limit model for L, a model M forgates to a limit model for L, a model M for L LP,ωP,ω..

R eferences K e i s l e r , H .J ., M o d e l T h e o ry fo r In f i n i t a ry L o g i c , N o r t h H o l l a n d , A m s t e rd a m , 1 9 7 1 .N o u ra n i ,C .F .," F u n c t o r i a l M o d e l T h e o ry a n d In f i n i t e L a n g u a g e C a t e g o r i e s ," S e p t e m b e r 1 9 9 4 , P re s e n t e d t o t h e A S L , J a n u a ry 1 9 9 5 , S a n F ra n c i s c o .[A S L B u l l e t i n s 1 9 9 6 ] . N o u ra n i , C .F . 2 0 0 5 a “ F rg a m e n t C o n s i s t e n t A l g e b ra i c M o d e l s , J u l y 2 0 0 6 C a t e g o r i e s O k t o b e r fe s t , U O t t a w a , O c t o b e r 2 0 0 5 N o u ra n i ,C .F . 2 0 0 5 b “ P o s i t i v e C a t e g o r i e s a n d P ro c e s s A l g e b ra s , ” D ra f t o u t l i n e A u g u s t 2 0 0 5 , W ri t t e n t o a S t a n fo rd C S e x p l o ra t i o n p ro j e c t .N o u ra n i ,C .F .,” F u n c t o r i a l G e n e r i c F i l t e rs ,” J u l y 2 0 0 5 , A S L A b s t ra t c s 2 0 0 6 , M o n t re a l , C a n a d a . N o u ra n i ,C .F ,” F u n c t o r i a l S t r i n g M o d e l s ,” E R L O G O L -2 0 0 5 : In t e rm e d i a t e P ro b l e m s o f M o d e l T h e o ry a n d U n i v e rs a l A l g e b ra , J u n e 2 6- - J u l y 1 , S t a t e T e c h n i c a l U n i v e rs i t y / M a t h e m a t i c s In s t i t u t e , N o v o s i b i r s k , R u s s i a . w w w .n s t u .ru / s c i e n c e / c o n f / e r l o g o l- 2 0 0 5 .p h t m l, w w w .a m s .o rg / m a t h c a l / i n fo /2 0 0 5 _ j u n 2 6- j u l 1 _ n o v o s i b i r s k .h t m l N o u ra n i , C .F . 2 0 0 5 a “ F rg a m e n t C o n s i s t e n t A l g e b ra i c M o d e l s , J u l y 2 0 0 6 C a t e g o r i e s O k t o b e r fe s t , U O t t a w a , O c t o b e r 2 0 0 5 N o u ra n i ,C .F . 2 0 0 5 b “ P o s i t i v e C a t e g o r i e s a n d P ro c e s s A l g e b ra s , ” D ra f t o u t l i n e A u g u s t 2 0 0 5 , W ri t t e n t o a S t a n fo rd C S e x p l o ra t i o n p ro j eP o s i t i v e R e a l i z a b i l i t y M o rp h i s m s a n d T a rs k i M o d e l s C y ru s F N o u ra n i . . .

w w w .m a t h .w i s c .e d u / ~ l e m p p / c o n f / .. . / n o u ra n i .p d f - D i e s e S e i t e ü b e rs e t z e n D a t e i fo rm a t : P D F / A d o b e A c ro b a t - S c h n e l l a n s i c h t P o s i t i v e R e a l i z a b i l i t y M o rp h i s m s a n d T a rs k i M o d e l s . C y ru s F N o u ra n i * . M a rc h 2 0 0 7 . P o s i t i v e R e a l i z a b i l i t y o n H o rn F i l t e rs w w w .l c 0 8 .i a m .u n i b e .c h / . . ./ C y ru s % 2 0 F 0 7 2 1 1 1 3.. . - D i e s e S e i t e ü b e rs e t z e n D a t e i fo rm a t : P D F / A d o b e A c ro b a t - S c h n e l l a n s i c h t

A c a d e m i a C l i fo rn i a a n d B e r l i n A k d m k rd .t r i p o .c o m

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