Melvin's digital garden

Set theory

speaker: Desmond Lau event: SIC #8 ** History Dirichlet:

  • functions should be any correspondence between numbers,
  • not only those that can be expressed analytically Riemann
  • 1854: first mention of sets? Dedekind
  • 1871: founded algebraic number theory
  • 1872: constructed the reals using cuts of rationals Cantor
  • 1872: constructed the reals using Cauchy sequence of rationals

reals are subsets of rationals with certain properties, requires power set axiom ** Cantor’s contributions construction of reals cardinality diagonal argument cantor-bendixson theorem

** Cantor-Bendixson theorem ordinals general topology ** Ordinals = isomorphism classes of well-orders

well-order is a linear order where every subset has a least element isomorphism between two well-orders is a bijective function f where f(x) < f(y) => x < y

** General topology (1920s) (X,T), T collects subsets of X

  1. empty set is in T
  2. X is in T
  3. a,b in T => a intersect b in T
  4. union of any subset of T is also in T

members of T are the open sets of X closed sets are complements of open sets

** Limit point A is a subset of X x is a limit point of A <=> x in open set U => U intersect A-x is non-empty

eg sequence that converges to sqrt(2), sqrt(2) is the limit point

Claim: A is open <=> for all x in A, there exists U in T such that U subset of A and x in U Lemma 1: A is closed <=> the set of limit points of A is a subset of A Lemma 2: If A is a set of limit points of B, then a limit point of A is a limit point of B

** Cantor-Bendixson rank A subset of X A^0 = A A^n+1 = limit points of A^n A^l = intersection of A^x where x < l and l is limit

example of limit is w = infinity (size of natural numbers), note that w is not the successor of any natural number Cantor is the first to consider w and beyond

Lemma 3: If n > 0, then A^n = intersection of limit points of A^x for all x < n implies for all 0 < a < b, then A^a contains A^b if A is closed, for all a < b, A^a contains A^b (due to Lemma 1)

The Cantor-Bendixon rank of A is the least a such that A^a = A^a+1

** Cantor-Bendixson theorem if A is closed in R, then A is the disjoint union of B and C where B is countable and C = C’ (C is perfect) The standard topology of R is the set of open intervals

Let a be the rank of A, we want to show that A - A^a is countable.

Let x be an element of A - A^a Let b be the first step where x is taken out, i.e. x in A^b but not in A^b+1 x is not a limit point of A^b Let u_x be an open interval in R with rational endpoints where A^b intersect u_x = {x} If x,y in A - A^a, then x != y => u_x != u_y There are countably many open intervals with rational endpoint

** Modern set theory, ZF Axioms 0. Existence

  • there exists a set X
  1. Extensionality
  • two sets are equal if they have exactly the same members
  1. Schema of comprehension
  • for each formula p and set X, {x in X : p(x)} is a set
  1. Foundation
  • \in is well founded, there is no infinitely decreasing sequence of \in relations
  • implies no set is a member of itself
  1. Pairing
  • a,b are sets => {a, b} is a set
  1. Union
  • x is a set, union of elements of x is a set
  1. Replacement schema (Frankel)
  • for a set x and a function f on x, then range of f is a set
  1. Infinity
  • there exists a set X, such that empty set in X, y in X => S(y) in X
  • where S(y) = y union {y}
  1. Power set
  • every set has a power set
  1. Choice
  • if you have a family of non-empty sets F, there is a set that contains
  • exactly 1 member of each set in the family

** ordinals (von Neumann) an ordinal is a set that is transitive (b \in a \in A, then b \in A) well ordered by \in

Fact 1: Every set of ordinals is well-ordered by \in Fact 2: If A is an ordinal, then A = {x subset of A where x is an ordinal}

** cardinal a is a cardinal <=> |a| = a

for an ordinal a, |a| = least ordinal b st there is a bijection from a to b

w = |N| has the same cardinality as w+1, w+w, w^w

continuum hypothesis: w_1 = |power set of N| ?

Links to this note