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Game Normal Formulas in Model Theory: Equivalence and Strategy

This document explores the nuances of model theory, focusing on game normal formulas and their implications on equivalence relations in first-order logic. It establishes the conditions under which two structures possess winning strategies in a defined game, marked by the equivalence relation A ≈k B. Key theorems such as the Compactness Theorem and the Upward Löwenheim-Skolem Theorem are examined, demonstrating how these principles apply to finite and infinite models. The findings reveal foundational insights into the structure and behavior of logical systems.

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Game Normal Formulas in Model Theory: Equivalence and Strategy

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  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

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