Set Theory

Consultations by email arrangement. Use "Set Theory" in subjects of emails.

• Lecture: Wed 12:20, S4
• Tutorials: Andres Aranda, web
• Czech Class: Honza Kyncl, web

Topics covered

Feb 21

• Brief history of proofs in mathematics (Pythagoras theorem and its visual proof, Euclid's elements and a gap in proof of Proposition I). Not required for exam.
• Why do we want to study set theory? We want solid foundation to mathematics to resolve some problems:
  • Russel's paradox: set X is extraordinary if X ∈ X, ordinary otherwise. Let O be the set of all ordinary sets. Is O ∈ P?
  • Russel's paradox in popular language: "There is one barber in a town. He shaves exactly those men, who do not shave themselves. Does he shave himself?"
  • Berry paradox: "X is the smallest positive integer not definable in under sixty letters." Arguably X both exists and does not exist.
  • There are also highly counter-intuitive but correct results such at the Banach-Tarski paradox on sphere doubling.
  In 1900 these problems culminated thanks to the Hilbert's programme and led to the foundational crisis.
• Brief history: Bolzano introduced set theory to study infinity. Cantor's work initiated modern set theory. Russel-Whitehead: Principia mathematica and its proof that 1+1=2 as an attempt to resolve the foundational crisis, Goedel and his incompleteness results (very briefly)
• Brief crash-course on first order formulas. We will be most of time informal and use English, however we keep in mind that the underlying language is the predicate logic and everything we do can be expressed by it.
• Axiomatic approach: we will follow Zermelo--Fraenkel axiomatization from 1908 (Van Neumann-Gödel-Bernays is other popular)
  • The Axiom of Existence (Axiom existence): empty set exists. We found way to express it as formula
  • The Axiom of Extensionality (Axiom extenzionality): If two sets have precisely same elements they are equal. Again we wrote a formula
  • Thorem: Empty set is unique. Proof.

Feb 28

• Axiom scheme of (restircted) comprehension
• The Axiom of Pair -- "exists {A, B}"
  • for every A,B ex. C s.t. x ∈ C iff x=A or x=B
  • Notation {A,B} or {A} -- can be used, as it is unique.
  • the first axiom that provides more than an empty set!
• The Axiom of Union -- for every S ex. U s.t. x ∈ U iff x ∈ A for some A ∈ S
  • notation \(U = \cup S\) (usually \(\cup_{A \in S} A\))
  • examples ... in particular union of two sets
• The Axiom of Power Set ⊆ -- a shortcut for subset0 for every S ex. P s.t. X ∈ P iff X ⊆ S
  • examples:
  • Elementary operations with sets: union, symmetric difference, set minus,.... Proofs that those are also sets.
  • Intersection of empty set is an empty set
• Ordered pair (due to Kuratowski): \((a,b)=\{a,\{a,b\}\}\)

  This is a smart definition which makes it possible to define relations and functions and prove basic facts. Ordered n-tuples are also easy: (a,b,c)=((a,b),c).

  • Relations (following what we know from discrete mathematics). Dom R, ran R, field R all of them are sets because they are subsets of \(\cup \cup R\)
  • Image, preimage
  • Inverse relations: we need scalar product to show it is a set.
  • image under inverse relation is the inverse image

Feb 28

Functions
• Compatible functions
• Theorem: union of compatible functions is a function.

Building \(\mathbb N\):
• \(0 = \{\}, 1 = \{0\}, \ldots , 17 = \{0,1,..., 16\}\)
• This is another smart definition. It makes it possible to define successor operation: \(S(x)=x\cup \{x\}\). We will also write \(x+1\) for \(S(x)\).
• inductive set \(I\): \(0 \in I\) and \(x \in I \implies S(x) \in I\)
• Axiom of Infinity: An inductive set exists.

  (We need new axiom here: this does not follow from what we already have)

• Def: The set of all natural numbers is defined by \(\mathbb N= \{x | x \in I\) for every inductive set \(I\}\).
• This definition is legal: \(\mathbb N\) is a set because it is a subset of any given inductive set and is unique.
• \(\mathbb N\) is inductive and it is a subset of every inductive set
• Def ordering on \(\mathbb N\) by \(\in\)
• The induction principle : P(x) a property (a formula with free variable x), assume
  • P(0)
  • \(\forall n \in N: P(n) \implies P(n+1)\)
  Then P(n) holds for all \(n \in \mathbb N\)

  Proof: The set \(A = \{n \in N : P(n) \}\) is inductive

• Thm \((\mathbb N, <)\) is a linearly ordered set

  To prove this we used induction principle to show transitivity, assymetricity. Hard step was linearity that needed double induction.

Literature

Many recommended books are mentioned in the SIS description of the class. We loosely follow the book by Hrbacek and Jech. You may want to consult notes of Václav Končický (in Czech).