Set Theory Running Notes.md

Foundations

• Formal system in math?
  • Definition
  • Components
  • Needs
• What is the difference between a definition and a proposition in a formal language?
• Difference between semantics and syntax?

Set Theory

History

• How were sets used in mathematical language before cantor and introduction of infinite sets?
• What were the paradoxes caused by considering infinite sets to be worth studying?
  • Difference between paradoxes caused by real vs virtual infinity?

Defining every class/sets

• Which Objects can be defined by sets
  • Ordered N-Tuples
    • Groups/Fields/Rings (Ordered Pair of a set and a binary operator relation)
    • Sequences, Real Numbers, Infinite Sums, Infinite Radicals and Fractions
  • Graphs
    • Functions, Relations
  • Power Sets
    • Ways of choosing from a bigger set (combinatorics)
• Which objects cant be defined by sets/can be defined by proper class?
  • Class of all sets/classes
  • Class of all groups
  • Class of all vector spaces
  • Surreal numbers
  • Ordinal Numbers

Axiomatic Set Theory Outline

• Primitive Notions
  • Set/Class for NBG
    • What is the difference between Set, Class, Proper Class
  • Elementhood
• Axioms
  • Extentionality
  • Specification (for )
  • Subclass
  • Pair
  • Power Set
  • Union
  • Regularity (restrict to well-founded classes only)
  • Replacement (Mapping from set gives set)
• Definitions:
  • Equality (some times treated as a primitive)
    • Difference between extensionality axiom vs equality definition vs equality in logic
  • Subset
  • Union/Intersection
  • Ordered N-tuple/pair
  • Cartesian Product/Graph
  • Generalized Union/Intersection

Types of Classes

• What are sets?
• What are classes?
• What are proper classes?
• What are urelements?
• Can cartesian product of two sets be a proper class?
  • Do same for other types of big sets:
    • Union/Infinite Union
    • Mapping?
    • Power set?

Extensionality

• Difference between extensionality and intensionality?
• Difference between axiom of extension vs equality as a primitive

Specification

• What are syntactic paradoxes?
• What are semantic paradoxes?
• How does ZFC deal with Russel's Paradox?
• How does NBG deal with Russel's Paradox?

Replacement

Building Sets/Classes

• NBG comprehension: Build set based on property