1 / 19

Model Theory

Max Euwe. Model Theory. Jouko Väänänen. Lemma. Denote A  k B if  has a winning strategy in EF k [A,B]. TFAE (A, a )  k+1 (B, b ) For every cA, then there is dB such that (A, a c)  k (B, b d), and for every dB there is cA such that (A, a c)  k (B, b d). Game normal formulas.

archie
Download Presentation

Model Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Max Euwe Model Theory Jouko Väänänen Model theory

  2. Lemma • Denote AkB if  has a winning strategy in EFk[A,B]. • TFAE • (A,a) k+1(B,b) • For every cA, then there is dB such that (A,ac) k(B,bd), and for every dB there is cA such that (A,ac) k(B,bd). Model theory

  3. Game normal formulas • Let 0,…,m-1 be all the unnested atomic formulas in x0,…,xn-1. We assume the signature is finite. • Let n,0 be the finite set of all consistent 0s(0)…m-1s(m-1) where s:m{0,1} and 1= , 0= . • Let 0,…,j-1 be all the finitely many formulas in x0,…,xn inn+1,k • Let n,k+1be the finite set of all iXxni(x0,…,xn)xniXi(x0,…,xn) where X{0,…,j-1}. • Elements of n,k are called game normal formulas. Model theory

  4. Formulas in n,k are mutually contradictory. • k=0: Clear by definition. • Suppose AiXxni(a0,…,an-1,xn)xniXi(a0,…,an-1,xn) AkYxnk(a0,…,an-1,xn)xnkYk(a0,…,an-1,xn) • We show X=Y. If iX, then Ai(a0,…,an) for some an. Hence Ak(a0,…,an) for some kY. By Induction Hypothesis, i=k. Converse is similar. Model theory

  5. Formulas in n,k cover all cases. • A any model, a a sequence a0,…,an-1 in A. • Claim: There is some (x0,…,xn-1) in n,k such that A(a0,…,an-1). • k=0: clear. • k+1: Let 0,…,j-1 be all the finitely many formulas in x0,…,xn inn+1,k. Let X={i:Ai(a0,…,an-1,b) for some b}. • ThenAiXxni(a0,…,an-1,xn)xniXi(a0,…,an-1,xn). Model theory

  6. Formulas in n,k describe winning positions. TFAE • (A,a) k(B,b) • a satisfies the same formula in n,k as b in B. True if k=0. Assume (A,a) k+1(B,b) and AiXxni(a,xn)xniXi(a,xn). We show BiXxni(b,xn)xniXi(b,xn). Fix i. There is c in A such that Ai(a,c). By assumption, there is d in B such that (A,ac) k(B,bd). By Induction Hypothesis, Bi(b,d). Fix then d in B such that for some i Bi(b,d). Again by assumption, there is c in A such that (A,ac) k(B,bd). By Induction Hypothesis, Ai(a,c). Assume then (2), e.g. AiXxni(a,xn)xniXi(a,xn) and BiXxni(b,xn)xniXi(b,xn). To prove (A,a) k+1(B,b), let c in A. There is i such that Ai(a,c). Hence there is d such that Bi(b,d). By Induction Hupothesis, (A,ac) k(B,bd), as desired. Similarly if d in B and Bi(b,d), there is c in A such that Ai(a,c), and again (A,ac) k(B,bd). QED

  7. Every unnested formula of quantifier rank k is equivalent to a disjunction of formulas in n,k • Let (x0,…,xn-1) have quantifier rank k. • Let 0,…,m-1 be all the finitely many formulas in x0,…,xn inn,k that are consistent with (x0,…,xn-1). • Claim: For all A and a: A (a)i i(a). • “” is clear, because n,k “covers all cases”. • “” Suppose Ai(a). There are B and b such that B (b)i(b). (A,a) k(B,b). So A (a). QED Model theory

  8. Equivalence relation AnB i.e. the same sentence of 0,n is true in A and B Only finitely many classes, because each 0,n is finite Lecture 3

  9. n - classes of models Each equivalence class is definable by a sentence of 0,n

  10. n - classes Every model class which is definable by a sentence of quantifier rank  n, is a union of equivalence classes

  11. A game characterization of first order logic A model class is first order definable if and only if For somen the model class is closed under n Lecture 3

  12. Game normal form • Every first order formula (x0,...,xn-1) of quantifier rank at most k is equivalent to a disjunction of a finite number of formulas in n,k. Model theory

  13. Not first order definable M is finite Infinite At least n elements n Lecture 3

  14. Not first order definable Among finite models: M has even number of elements 2n+1 elements 2n elenemts n Lecture 3

  15. Not first order definable Lecture 3

  16. Compactness • Compactness Theorem: Let T be a first order theory. If every finite subset of T has a model, then so does T. • Proof: Add constant symbols, as many as there are formulas. Extend T to a Hintikka set T’. T’ has a model A’. The reduct A of A’ to the original signature is a model of T. Model theory

  17. Upward Löwenheim-Skolem Theorem • Suppose A is a model of size   in a signature of size  . Then A has an elementary extension of size . • Add names to all elements of A • Let T=eldiag(A){cicj: i<k<}. • By compactness T has a model B of cardinality  . W.l.o.g. A≺ B. • By Downward Löwenheim-Skolem Theorem there is C of cardinality  such that A≺C≺B. Model theory

  18. On the number of models • Let T be a countable complete first order theory. • Let nT() be the number of models of T of cardinality  (up to isomorphism). • Löwenheim-Skolem Theorem:nT()>0 for some infinite  if and only if nT()>0 for all infinite . • Morley’s Theorem: nT()=1 for some uncountable  if and only if nT()=1 for all uncountable . • Shelah’s Theorem:If , then nT()nT(). • Shelah’s Main Gap: For all T,nT() is either very slowly growing (sc. structure case) or very fast growing (sc non-structure case) Model theory

  19. Vaught’s Conjecture • Vaught Conjecture says: If T is countable complete first order theory then nT() or nT()=2. Model theory

More Related