Title: Daniel's Flashcards / Year 4 / Term 2 / Model Theory

html: Definition 5.6. Let ( A j ) j ∈ J, U (ultrafilter on J), ∼ as before, and let n ∈ ℕ.
For each j ∈ J, suppose B j ⊆ A j n. Define
[ ( B j ) j ∈ J ] = { ( [ ( a j 1 ) j ∈ J ] , … , [ ( a j n ) j ∈ J ] ) ∈ ( ∏ j ∈ J A j ∕ U ) n : { j ∈ J : ( a j 1 , … , a j n ) ∈ B j } ∈ U } ⊆ ( ∏ j ∈ J A j ∕ U ) n

html: Definition 5.7. Let n > 0, k ∈ { 1 , … , n }. Define:
- π k ( x 1 , … , x n ) = ( x 1 , … , x k − 1 , x k + 1 , … , x n ).
- For X a set of n-tuples,
π k ( X ) = { ( x 1 , … , x k − 1 , x k + 1 , … , x n ) : ( x 1 , … , x n ) ∈ X } .

html: Definition 5.9. Suppose A j = A for each j ∈ J. Then we call ∏ ⁡ j ∈ J A j ∕ U an ultrapower of A, and write A U.
If B ⊆ A, we write [ B ] for [ ( B ) j ∈ J ].

html: Definition 15.1. Let M be an infinite L-structure, and κ > | L | + ℵ 0. We say M is κ-saturated if for any A ⊆ M with | A | < κ, every type in S n M ( A ) is realised in M for all n.

html: Definition 15.4. Define S n ( T ) = S n M ( ∅ ) for any / some M ⊨ T (because if M , N ⊨ T, then S n N ( ∅ ) = S n M ( ∅ ) as T h ( M ) = T h ( N ) = T).

html: Definition 16.1. We say p ∈ S n M ( A ) is isolated if it is an isolated point with respect to topology on S n M ( A ) (i.e. { p } is open).

html: Definition 16.8. For κ ≥ ℵ 0, let I ( T , κ ) be the number of models of T of size κ (modulo isomorphism).

html: Definition 17.1. Given κ ≥ | L | + ℵ 0, we say T is κ-stable if for any M ⊨ T, | M | = κ we have | S 1 ( M ) | = κ.
We say T is stable if it is κ-stable for some κ.

html: Definition 17.2. Let φ ( x , y ) be an L-formula, x , y types of finite length.
We say φ ( x , y ) has the order property with respect to T if there is some M ⊨ T, ( a i ) i ≥ 0, ( b j ) j ≥ 0 such that M ⊨ φ ( a i , b j ) if and only if i < j.

html: Definition 17.4. A theory T is strongly minimal if ∀ ⁡ M ⊨ T every definable subset of M is finite or cofinite.

html: Definition 17.5. Let M ⊨ T, A ⊆ M. Then b ∈ acl ⁡ ( A ) if there is an L A-formula ϕ ( x ) such that
M ⊨ ∃ ⁡ x = n ϕ ( x )
and M ⊨ ϕ ( b ).