0Review of First Order Logic
0.1Languages
0.2Structures
0.3Formulas / sentences
1Complete Theories
2Homomorphisms
3Categoricity
4Filters
5Ultraproducts
6Ultraproduct Structures
7Łoś’s Theorem and Consequences
8More Constructions
9Algebraically closed fields
10Diagrams
11Introduction to Quantifier Elimination
12Examples
13Introduction to Types
14Type Spaces
15Saturated Models
16Omitting Types
17Whistle Stop Tour of Stability Theory
Index

Introduction

Example. $(ℤ, +)$ is not the same as $(ℤ, +, \cdot)$ .

$(ℤ, +)$ is decidable (there exists an algorithm to decide whether a given sentence is true or false in the model).

However, $(ℤ, +, \cdot)$ is not decidable (Gödel’s completeness theorem).

Example.

(1) $(ℂ, +, 0, 1)$
(2) $(V, +, f_{r})$ (where $V$ is an $ℝ$ -vector space, and $f_{r}$ is the map $r : v \mapsto r v$ for any $r \in ℝ$ ).

These structures are both strongly minimal.

Definition (Strongly minimal). A theory is strongly minimal if all formulas in one variable are either finite or co-finite.

For the $ℂ$ example: formulas in one variable are polynomial equations or inequations, so solution set is always either finite or cofinite (recall Fundamental Theorem of Algebra).

For the vector space example: the formulas in one variable are of the form $a x = b$ or $a x \neq b$ .

Cheats:

Boolean combinations and quantifiers:
$\exists y a x y^{2} + y + x^{2} + 3 = 0 .$

Need quantifier elimination (boolean combinations are easy to deal with).
Elementary extensions (chapter 1).

Interestingly: strongly minimal structures all carry notion of dimensions. For example:

In $(ℂ, +, \cdot, 0, 1)$ this is transcendence degree.
In ${(V, +, f_{r})}_{r \in ℝ}$ , this is linear dimension.

If interested in further reading: see

https://forkinganddividing.com/

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Contents

Introduction