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

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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 x0, x1, . . .,

predicate symbols {Pi | i ∈ I} and constants {cj | j ∈ J}. I and J are usually finite, or at least countable.

• If needed, λ(i) gives the arity of predicate Ri
• 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.

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• Given an L-structure M, we say that L is the lan-

guage for M

• A sequence σ = a0, a1, . . . of elements of A is called

an assignment; ai is the value given to variable vi.

• we write σ(n | b) for a0, a1, . . . an−1, b, an+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 vi = vj iff ai = aj

• 2. M |

=σ c = vj iff aj = c for a constant c

• 3. M |

=σ Ri(t1, . . . , tλ(i)) iff (b1, . . . , bn) ∈ R(i) so that for each i M | =σ bi = ti

• 4. M |

=σ ∃viφ 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

• An embedding of M into M′ is a one-to-one map-

ping f : A → A′ such that

• 1. f(cj) = c′

j for all j ∈ J

• 2. (a1, . . . , an) ∈ Ri implies (f(a1), . . . , f(an)) ∈ 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

closed formula φ, M | = φ iff M′ | = φ

More preliminaries

• An elementary substructure of M is the structure

M′ such that M′ ⊆ M and M′ | = φ(a1, . . . , an) if and

• nly if M |

= φ(a1, . . . , an), for every a1, . . . , an ∈ A′

• 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(a1, . . . , an) iff M′ | = φ(f(a1), . . . , f(an)) for all a1, . . . , an.

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• 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′

• Embedding Theorem:

M ≺ M′ is equivalent to saying that for any n and L-formula φ with n + 1 free variables, and any n constants, if M′ | = φ(a1, . . . , an, a) for some a ∈ D′ then there is some a′ ∈ A such that M | = φ(a1, . . . , an, a′) – to prove implication: M′ | = φ(a1, . . . , an, a) im- plies M′ | = ∃x(φ(a1, . . . , an, x)), and thus M | = ∃x(φ(a1, . . . , an, x)).

– The other direction is by induction on the length

• f φ. Induction steps for ∧ and ¬ are trivial, and

the only trouble remains with ∃xψ (left as an exercise)

Indexing of sets

• Let LK be the language obtained from L by adding

constansts c′ = {ck | k ∈ K}

• Then the LK-structure Mc′ = (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 Mc′ is elementarily equivalent to Mc∗.

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

L¨

• wenheim-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′

• f 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.

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– Define B0, B1, . . . as follows: B0 is any set for which X ⊆ B0 and |B0| = α. – Given a formula φ with m free variables, Yn(φ) = {x ∈ A | ∃a1, . . . am−1 ∈ Bn(M | = φ(a1, . . . , am−1, x))} – Bn+1 = {h(Yn(φ)) | φ ∈ L} – This construction adds to Bn+1 elements of A that are needed to make formula in Bn with free variables true; and one for each such formula. – Bn = α for every n: |B0| = α and the there is at most α formulas in L, so Bn+1 ≤ α · α = α for all infinite cardinalities

– We define A′ = B0 ∪B1 ∪· · ·, and the Embedding theorem gives us that M′ ≺ M

• The Downward L¨
• wenheim 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 LU contains at most γ symbols, i.e., LU is the smallest language that can express U.

• Then, by the general theorem, U has a model with

cardinality |LU| = |N|

• Corollary: Any countable set of closed formulas has

a countable model.

• The Upward L¨
• wenheim-Skolem theorem: A set of

closed formulas U with a model |M| ≥ |N| has a model for every cardinality ≥ |M|

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

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• We assume we have a family of models, Mi =

(Ai, Ri) for i ∈ I and (A, S) = Πi∈IMi is the di- rect product of the whole family. In this case A is the cartesian product of all Ai and S = {(f, g) | (f(i), g(i)) ∈ Ri}

• The direct product is not very convenient on its
• wn, 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!)

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)) ∈ Ri}

• 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

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– [f = g] = E(f, g) – [R(f, g)] = R(f, g) – [σ1 ∧ σ2] = [σ1] ∩ [σ2] – [¬σ] = I \ [σ] – When x is a free variable in φ, we denote [∃xφ(x)] =

• f∈A

[σφ(f)]

• [·] assigns a “Boolean truth values” in the boolean

algebra of 2I 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

• f the models in the product actually satisfy the

formula

• Theorem: For a formula φ(x1, . . . , xn) we have

[φ(f1, . . . , fn)] = {i ∈ I | Mi | = φ(f1(i), . . . , fn(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 Xξ = [φ(fξ)]\

• η<ξ[φ(fη)], i.e., for the well-ordered set of con-

stants, we take the indices for which those “smaller than” a given fξ make the formula true.

– If α is an ordinal “big enough” then [∃xφ(x)] =

• η<α[φ(fη)]

– Because the product’s domains are disjoint, Xξ ∩ Xη = ∅ for ξ = η. We choose f ∈ A so that f | Xξ = fξ | Xξ 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) ∈ RF iff [R(f, g)] ∈ F

• Lemma: Let F be a filter over I. Then ∼F is an

equivalence relation on A, and: f ∼F f′ and g ∼F g′, and (f, g) ∈ RF imply that (f′, g′) ∈ RF – 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.

The ultraproduct

• Let F be an ultrafilter over I; intuitively, an ultra-

filter consists of “large” subsets of I, in the sense that its canonical homomorphism maps sets of I

• nly as either 0 or 1.
• In the ultrafilter - which corresponds to a complete

theory - f ∼F g asserts that f and g agree on a large subset if indices; they are “the same” from the point of view of the theory.

• For each f ∈ A let f/F be the equivalence class of

f under ∼F and denote A/F = Πi∈IAi/F = {f/F | f ∈ A}

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and R/F = {(f/F, g/F) | (f, g) ∈ RF}

• We call the structure ΠMi/F = (A/F, R/F) the ul-

traproduct of the family Mi

• Theorem: let φ(x1, . . . , xn) be a L-formula.

Then the ultraproduct ΠMi/F | = φ(f1/F, . . . , fn/F) iff [φ(f1, . . . , fn)] ∈ F

Los’ Theorem

• For any L-formula φ with n free variables and any

f1, . . . , fn ∈ ΠAi we have ΠMi/F | = φ(f1/F, . . . , fn/F) ≡ {i ∈ I | Mi | = φ(f1(i), . . . , fn(u))} ∈ F

• the theorem says that the ultraproduct induced by

F is a model of φ iff the set of indices such that Mi | = φ is in the ultrafilter F

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Compactness theorem

• Let U be a set of closed L-formulas. If each finite

subset of U has a model, then U has a model – Suppose every finite subset of U has a model; let I be the family of all finite subsets of U. – For each W ∈ I let MW be a model of W and define ˆ W = {W ′ ∈ I | W ⊆ W ′} For W1, . . . , Wn ∈ I, we clearly have W1 ∪ · · · Wn ⊆ ˆ W1 ∩ · · · ∩ ˆ Wn; so that the set { ˆ W | W ∈ I} has the fmp in the boolean algebra of subsets of I – Let F be an ultrafilter over I containing each ˆ W.

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– Then the ultraproduct ΠW∈IMW/F is a model for U: Suppose σ ∈ U. Then for W = {σ}, MW | = σ. – Therefore ˆ {σ} = {W ∈ I | {σ} ⊆ W} ⊆ {W ∈ I | MW | = σ} ∈ F – The Los’ Theorem then implies ΠW∈IMW/F | = σ This holds for every σ, so this completes the proof.