lecture slides for mat 60556 part viii model theory
play

Lecture Slides for MAT-60556 Part VIII: Model theory Henri Hansen - PowerPoint PPT Presentation

Lecture Slides for MAT-60556 Part VIII: Model theory Henri Hansen November 5, 2013 1 Preliminaries We denote by L a first-order logic with strict equal- ity (=) and no functions (but with constants) We assume a countable set of


  1. Lecture Slides for MAT-60556 Part VIII: Model theory Henri Hansen November 5, 2013 1

  2. Preliminaries • We denote by L a first-order logic with strict equal- ity (=) and no functions (but with constants) • We assume a countable set of variables x 0 , x 1 , . . . , predicate symbols { P i | i ∈ I } and constants { c j | j ∈ J } . I and J are usually finite, or at least countable. • If needed, λ ( i ) gives the arity of predicate R i • An L -structure is a triple M = ( A, R , c ) where R is a mapping from I to the set of relations on A , and c is a mapping J �→ A . 2

  3. • Given an L -structure M , we say that L is the lan- guage for M • A sequence σ = a 0 , a 1 , . . . of elements of A is called an assignment; a i is the value given to variable v i . • we write σ ( n | b ) for a 0 , a 1 , . . . a n − 1 , b, a n +1 , . . . , i.e., b is put into the n:th place of σ • We define the satisfying relation, M | = σ φ for for- mulas of L inductively as follows: 1. M | = σ v i = v j iff a i = a j

  4. 2. M | = σ c = v j iff a j = c for a constant c 3. M | = σ R i ( t 1 , . . . , t λ ( i ) ) iff ( b 1 , . . . , b n ) ∈ R ( i ) so that for each i M | = σ b i = t i 4. M | = σ ∃ v i φ iff M | = σ ( i | b φ for some b ∈ A • If φ has no free variables (is a closed formula), we write simply M | = φ if M | = σ φ for some σ • M ′ = ( A ′ , R ′ , c ′ ) is a substructure of M iff A ′ ⊆ A and R ′ ( i ) = R ( i ) ∩ A ′ λ ( i ) for each i • A restriction of M into a set B ⊆ A is denoted M | B , it is defined in the natural way; we require that the image of c is contained in B

  5. • An embedding of M into M ′ is a one-to-one map- ping f : A �→ A ′ such that 1. f ( c j ) = c ′ j for all j ∈ J 2. ( a 1 , . . . , a n ) ∈ R i implies ( f ( a 1 ) , . . . , f ( a n )) ∈ R ′ i • If an embedding is a bijection, then it is an isomor- phism . If an isomorphism exists we write M ∼ = M ′ • For an arbitrary embedding f , f [ M ] ∼ = M • M and M ′ are elementarily equivalent iff for every = φ iff M ′ | closed formula φ , M | = φ

  6. More preliminaries • An elementary substructure of M is the structure M ′ such that M ′ ⊆ M and M ′ | = φ ( a 1 , . . . , a n ) if and = φ ( a 1 , . . . , a n ), for every a 1 , . . . , a n ∈ A ′ only if M | • If M ′ is an elementary substructure of M , we write M ′ ≺ M • M ∼ = M ′ does not imply M ≺ M ′ !! • An embedding of M into M ′ is an elementary em- bedding iff for any formula φ we have: M | = φ ( a 1 , . . . , a n ) iff M ′ | = φ ( f ( a 1 ) , . . . , f ( a n )) for all a 1 , . . . , a n . 3

  7. • M is elementarily embeddable to M ′ if there is an elementary embedding from M to M ′ • M is elementary embeddable to M ′ if M is isomor- phic to some elementary substructure of M ′ M ≺ M ′ is equivalent to • Embedding Theorem: saying that for any n and L -formula φ with n + 1 free variables, and any n constants, if M ′ | = φ ( a 1 , . . . , a n , a ) for some a ∈ D ′ then there is some a ′ ∈ A such that M | = φ ( a 1 , . . . , a n , a ′ ) M ′ | – to prove implication: = φ ( a 1 , . . . , a n , a ) im- plies M ′ | = ∃ x ( φ ( a 1 , . . . , a n , x )), and thus M | = ∃ x ( φ ( a 1 , . . . , a n , x )).

  8. – The other direction is by induction on the length of φ . Induction steps for ∧ and ¬ are trivial, and the only trouble remains with ∃ xψ (left as an exercise)

  9. Indexing of sets • Let L K be the language obtained from L by adding constansts c ′ = { c k | k ∈ K } • Then the L K -structure M c ′ = ( A, R , c ∪ c ′ ) is an ex- pansion of ( A, R , c ). If c ′ maps K surjectively to A , then we say that c ′ is an indexing of A by K • Lemma: Let M and M ′ be L -structures, and let c ′ be an indexing of A by K . Then M is elementarily embeddable to M ′ iff there is a mapping c ∗ : K �→ A ′ such that M c ′ is elementarily equivalent to M c ∗ . 4

  10. – Given an elementary embedding f , c ∗ ( k ) = f ( c ′ ( k )) results in M c ∗ ≡ M c ′ – Given c ∗ : K �→ A , define f ( c ′ ( k )) = c ∗ ( k ) results in f being an elementary embedding

  11. L¨ owenheim-Skolem Theorems • We denote by | M | the cardinality of the model (its domain). • General Theorem: Let M be an infinite L -structure, and X ⊆ A Then for any cardinal α ≤ | M | , α ≥ | X | , and α ≥ |L| there is an elementary substructure M ′ of M such that | M ′ | = α and X ⊆ A ′ . – Let h be a choice function for the non-empty subsets of A (use axiom of choice), i.e., h ( Y ) ∈ Y for every Y ⊆ A that is not empty. 5

  12. – Define B 0 , B 1 , . . . as follows: B 0 is any set for which X ⊆ B 0 and | B 0 | = α . – Given a formula φ with m free variables, Y n ( φ ) = { x ∈ A | ∃ a 1 , . . . a m − 1 ∈ B n ( M | = φ ( a 1 , . . . , a m − 1 , x )) } – B n +1 = { h ( Y n ( φ )) | φ ∈ L} – This construction adds to B n +1 elements of A that are needed to make formula in B n with free variables true; and one for each such formula. – B n = α for every n : | B 0 | = α and the there is at most α formulas in L , so B n +1 ≤ α · α = α for all infinite cardinalities

  13. – We define A ′ = B 0 ∪ B 1 ∪· · · , and the Embedding theorem gives us that M ′ ≺ M • The Downward L¨ owenheim Skolem theorem: Let U be a set of closed formulas of L and let U have a model of cardinality α . Then U has a model for all the cardinalities β s.t. max( | U | , | N | ) ≤ β ≤ α • Let M be the model with cardinality α . Let γ be the cardinality of | U | (or | N | which ever is larger). Then L U contains at most γ symbols, i.e., L U is the smallest language that can express U . • Then, by the general theorem, U has a model with cardinality |L U | = | N |

  14. • Corollary: Any countable set of closed formulas has a countable model. • The Upward L¨ owenheim-Skolem theorem: A set of closed formulas U with a model | M | ≥ | N | has a model for every cardinality ≥ | M |

  15. From Ultraproducts to Compactness • For proof-technical reasons, we restrict, for this part, our attention to models that only have the domain and a single binary predicate, i.e. the model is simply of the form( A, R ) • The language L here is a first-order language that uses this predicate and no constants. The exten- sion of this proof to full first-order logic is straight- forward, but the technicalities quickly become ex- tremely tedious. Please try to think at each point, how the concepts would translate to arbitrary pred- icates with constants and functions 6

  16. • We assume we have a family of models, M i = ( A i , R i ) for i ∈ I and ( A, S ) = Π i ∈ I M i is the di- rect product of the whole family. In this case A is the cartesian product of all A i and S = { ( f, g ) | ( f ( i ) , g ( i )) ∈ R i } • The direct product is not very convenient on its own, because the direct product does not share first-order properties of its components. For in- stance, ∀ x ∀ y ( R ( x, y ) ∨ R ( y, x )) may hold for each member, but not for the prouct (show the exam- ple!)

  17. Modified product • We define two mappings, E and R for the product as follows – E ( f, g ) = { i ∈ I | f ( i ) = g ( i ) } – R ( f, g ) = { i ∈ I | ( f ( i ) , g ( i )) ∈ R i } • Let L ( A ) be the language obtained from L by adding a constant symbol for each f ∈ A • We define a mapping [] for closed formulas of L ( A ) into sets of indices inductively as follows 7

  18. – [ f = g ] = E ( f, g ) – [ R ( f, g )] = R ( f, g ) – [ σ 1 ∧ σ 2 ] = [ σ 1 ] ∩ [ σ 2 ] – [ ¬ σ ] = I \ [ σ ] – When x is a free variable in φ , we denote [ ∃ xφ ( x )] = � [ σφ ( f )] f ∈ A • [ · ] assigns a “Boolean truth values” in the boolean algebra of 2 I to sentences if L ( A ), so that the “truth value” of a formula is now a set of indices instead of true or false. The indeces denote which

  19. of the models in the product actually satisfy the formula • Theorem: For a formula φ ( x 1 , . . . , x n ) we have [ φ ( f 1 , . . . , f n )] = { i ∈ I | M i | = φ ( f 1 ( i ) , . . . , f n ( i )) } The proof is by induction, left as an exercise • Lemma: Let φ ( x ) be a formula of L ( A ). Then there exists and f ∈ A such that [ ∃ xφ ( x )] = [ φ ( f )] – Let < be a well-order for A and let X ξ = [ φ ( f ξ )] \ η<ξ [ φ ( f η )], i.e., for the well-ordered set of con- � stants, we take the indices for which those “smaller than” a given f ξ make the formula true.

  20. – If α is an ordinal “big enough” then [ ∃ xφ ( x )] = η<α [ φ ( f η )] � – Because the product’s domains are disjoint, X ξ ∩ X η = ∅ for ξ � = η . We choose f ∈ A so that f | X ξ = f ξ | X ξ and we can deduce that for this f [ ∃ xφ ( x )] = [ φ ( f )] • Let F be some collection of subsets of I . We define – f ∼ F g iff [ f = g ] ∈ F – ( f, g ) ∈ R F iff [ R ( f, g )] ∈ F • Lemma: Let F be a filter over I . Then ∼ F is an

  21. equivalence relation on A , and: f ∼ F f ′ and g ∼ F g ′ , and ( f, g ) ∈ R F imply that ( f ′ , g ′ ) ∈ R F – Recall that [ f = g ] denotes the set { i ∈ I | f ( i ) = g ( i ) } – We need to show that ∼ F is transitive, i.e., that [ f = g ] ∈ F and [ g = h ] ∈ F imply that [ f = h ] ∈ F . – The meet in I is intersection so [ f = g ] ∩ [ g = h ] ∈ F , and it is clear that [ f = g ] ∩ [ g = h ] ⊆ [ f = h ] and because ⊆ is the partial order of F the result follows.

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend


More recommend