Hilbert s Nustellensatz: Inner Model Theory: Ahmed T Khalil. Elliot Glazer. REU Advisor: Dr. Grigor Sargsyan. A Model-Theoretic Approach

Transcription

1 Inner Model Theory: An Introduction Hilbert s Nustellensatz: A Model-Theoretic Approach Elliot Glazer Ahmed T Khalil REU Advisor: Dr. Grigor Sargsyan

2 Part One Inner Model Theory

3 Models of ZFC ZFC is the most popular axiomatization of set theory, and widely considered the foundation of mathematics Allows for basic set theoretic operations like union and replacement ZFC takes place in the universe of sets, V, aka the Von Neumann universe However, there are subclasses of V that also satisfy the axioms of ZFC Studying these other models of ZFC gives us insight into the theory, including what can and cannot be proven by its axioms

4 What is an Inner Model? An inner model is a model M of ZFC that satisfies two important properties: 1. An inner model must be a proper class (i.e. it must not be an element of V) 2. It must be a transitive model, i.e. if S is an element of M, and T is an element of S, then T is an element of M Note: 2 implies that M knows what is inside each of its elements

5 Constructible universe The most important inner model of V is L, the constructible universe It is constructed by transfinite recursion by defining new sets from already constructed sets L is a model of ZFC, and in fact an inner model It is the least inner model of ZFC Satisfies the generalized continuum hypothesis It is consistent that V=L, i.e. the axiom of constructibility, but most set theorists don t like this axiom because L is a boring universe Regardless, L is very useful in set theoretic arguments

6 Back to determinacy Recall that analytic determinacy is the statement that all analytic sets are determined (i.e. there is a winning strategy in the corresponding game) Analytic determinacy follows from the existence of a measurable cardinal Existence of a measurable cardinal is actually equivalent to there being an inner model M and a non-trivial elementary embedding of V into M In such an embedding, the critical point of the embedding (the first ordinal moved) is necessarily a measurable cardinal

7 Part Two Hilbert s Nullstellensatz

8 The Saga Continues Recall from last time that we defined a language as follows: Definition: (Language) A language is a quadruple L = C, F, R, a where C is a set of constant symbols, F is a set of function symbols, R is a set of relation symbols, and a: F R N >0 is a function (arity function). In colloquial terms, a language is just a bunch of symbols. They are only meaningful once we have given these symbols meaning (defined an interpretation) over some structure (world) Definition: (L-structure) Let L be a language. A L-structure is a pair M = M, i where M is a set (universe of the structure) and i is a function such that if c C, then i c M if f F, then i f : M n M if r R, then i(r) M n

9 The Saga Continues A L-theory T is a collection of L-sentences These L-sentences are often called axioms Given any L -structure, you can then check if your structure satisfies your theory Example: 1) Establish language L =, e 2) Construct L-structure over some set M such that is interpreted as a binary function on M and e is interpreted as some constant in M 3) Check if this L-structure is a group (satisfies group axioms) A L-theory T is said to admit quantifier elimination if for every statement φ that has quantifiers in it, there is another statement ψ that has no quantifiers in it such T satisfies φ only when T also satisfies ψ φ and ψ may be completely unrelated A theory that admits quantifier elimination can be thought of as an unrich theory Showing that a theory admits quantifier elimination is pretty difficult

10 Humbling Hilbert As you ll recall from last time, Hilbert s Nullstellensatz is one of the most fundamental results in algebraic geometry Establishes a correspondence between the algebraic world and the geometric world Can be proven using model theoretic notions o Q: How? o A: Using Quantifier Elimination for ACF Again, it is generally very difficult to prove that a theory admits Q.E. due to existential nature of definition

11 The Timeline 1. Foundations of Mathematical Logic 2. Soundness & Completeness Theorems (Henkin Constructions) 3. Categoricity and Complete Theories 4. Quantifier Elimination for Dense Linear Orders (DLOs) 5. Quantifier Elimination for Divisible Abelian Groups (DAG) 6. Model Completeness 7. Quantifier Elimination for Algebraically Closed Fields 8. Foundations of Algebraic Geometry 9. Hilbert s Nullstellensatz

12 A Small Taste of the Fun The model theoretic proof of Hilbert s Nullstellensatz is contingent on the fact that ACF admits quantifier elimination The techniques used in the Q.E. proof for ACF are progressively built up in the proofs for Q.E. in DAG and Q.E. in DLO The easiest Q.E. proof is DLO

13 Post Research & Shoutouts Ideas Suggested by Dr. Sargsyan: compose a comprehensive set of lecture notes on Mathematical Logic & Model Theory create lecture videos on what I have learned continue reading under his guidance during the semester Special Thanks to: DIMACS REU Crew Dr. Grigor Sargsyan Elliot Glazer David Marker for his book Model Theory: An Introduction

14 Thank You