index of set theory
1 Basic Notions
- 1.
set theory
- 2.
set
- 3.
subset
- 4.
union
- 5.
power set
- 6.
generalized Cartesian product
- 7.
transitive set
- 8.
criterion for a set to be transitive
- 9.
Cartesian product
- 10.
proof of the associativity of the symmetric difference operator
- 11.
proper subset
- 12.
an example of mathematical induction
- 13.
principle of finite induction
- 14.
principle of finite induction proven from the well-ordering principle for natural numbers
- 15.
de Morgan’s laws
- 16.
de Morgan’s laws for sets (proof)
2 Functions and Relations
- 1.
antisymmetric
- 2.
example of antisymmetric
- 3.
argument
- 4.
constant function
- 5.
equivalence class
- 6.
direct image
- 7.
domain
- 8.
fibre
- 9.
fix (transformation
action)
- 10.
function
- 11.
function graph
- 12.
identity map
- 13.
inclusion mapping
- 14.
invariant
- 15.
inverse image
- 16.
irreflexive
- 17.
left function notation
- 18.
right function notation
- 19.
level set
- 20.
mapping
- 21.
mapping of period is a bijection
- 22.
operation
- 23.
operations on relations
- 24.
partial function
- 25.
partial mapping
- 26.
period of mapping
- 27.
properties of a function
- 28.
properties of functions
- 29.
quasi-inverse of a function
- 30.
range
- 31.
reflexive relation
- 32.
relation
- 33.
restriction
of a function
- 34.
set difference
- 35.
symmetric difference
- 36.
symmetric relation
- 37.
the inverse image commutes with set operations
- 38.
transformation
- 39.
transitive
- 40.
transitive closure
- 41.
transitive relation
- 42.
choice function
- 43.
one-to-one function from onto function
2.1 Order Relations
- 1.
poset
- 2.
maximal element
- 3.
minimal element
- 4.
visualizing maximal elements
- 5.
cofinality
- 6.
another definition of cofinality
- 7.
chain
- 8.
antichain
- 9.
branch
- 10.
tree (set theoretic)
- 11.
example of tree (set theoretic)
- 12.
proof that has the tree property
- 13.
filtration
- 14.
well ordered set
3 Cardinals and Ordinals
- 1.
-complete
- 2.
additively indecomposable
- 3.
aleph numbers
- 4.
algebraic numbers are countable
- 5.
all algebraic numbers
in a sequence
- 6.
another proof of cardinality of the rationals
- 7.
beth numbers
- 8.
Cantor normal form
- 9.
Cantor’s diagonal argument
- 10.
Cantor’s theorem
- 11.
cardinal arithmetic
- 12.
cardinal exponentiation under GCH
- 13.
cardinal number
- 14.
cardinal successor
- 15.
cardinality
- 16.
cardinality of a countable
union
- 17.
cardinality of disjoint union of finite sets
- 18.
cardinality of the continuum
- 19.
cardinality of the rationals
- 20.
classes of ordinals and enumerating functions
- 21.
club
- 22.
club filter
- 23.
countable
- 24.
countably infinite
- 25.
finite
- 26.
finite character
- 27.
fixed points of normal functions
- 28.
Fodor’s lemma
- 29.
Hilbert’s hotel
- 30.
if is infinite
and is a finite subset of then is infinite
- 31.
König’s theorem
- 32.
limit cardinal
- 33.
natural number
- 34.
normal (ordinal
) function
- 35.
open and closed intervals have the same cardinality
- 36.
ordinal arithmetic
- 37.
ordinal number
- 38.
pigeonhole principle
- 39.
proof of pigeonhole principle
- 40.
another proof of pigeonhole principle
- 41.
proof of Cantor’s theorem
- 42.
proof of fixed points of normal functions
- 43.
proof of Fodor’s lemma
- 44.
proof of the existence of transcendental numbers
- 45.
proof of theorems in additively indecomposable
- 46.
proof that countable unions are countable
- 47.
proof that the rationals are countable
- 48.
proof of Schroeder-Bernstein theorem
- 49.
Schroeder-Bernstein theorem
- 50.
stationary set
- 51.
subsets of countable sets are countable
- 52.
thin set
- 53.
successor
- 54.
successor cardinal
- 55.
the Cartesian product of a finite number of countable sets is countable
- 56.
law of trichotomy
- 57.
partitions less than cofinality
- 58.
Aronszajn tree
- 59.
example of Aronszajn tree
- 60.
Suslin tree
- 61.
Erdős-Rado theorem
- 62.
uncountable owned by yark
- 63.
uniqueness of cardinality
- 64.
Veblen function
- 65.
von Neumann integer
- 66.
von Neumann ordinal
- 67.
weakly compact cardinal
- 68.
weakly compact cardinals and the tree property
- 69.
inductive set
- 70.
inaccessible cardinals
4 Axiomatic Formulation
- 1.
axiom of choice
- 2.
axiom of countable choice
- 3.
axiom of determinacy
- 4.
axiom of extensionality
- 5.
axiom of infinity
- 6.
axiom of pairing
- 7.
axiom of power set
- 8.
axiom of union
- 9.
axiom schema of separation
- 10.
continuum hypothesis
- 11.
generalized continuum hypothesis
- 12.
equivalence of Zorn’s lemma and the axiom of choice
- 13.
Hausdorff
’s maximum principle
- 14.
Kuratowski’s lemma
- 15.
maximality principle
- 16.
permutation model
- 17.
Tukey’s lemma
- 18.
-small
- 19.
proof of Tukey’s lemma
- 20.
proof of Zermelo’s postulate
- 21.
proof of Zermelo’s well-ordering theorem
- 22.
proof that a relation is union of functions if and only if AC
- 23.
relation as union of functions
- 24.
Selector
- 25.
well-ordering principle for natural numbers proven from the principle of finite induction
- 26.
well-ordering principle implies axiom of choice
- 27.
Martin’s axiom
- 28.
Martin’s axiom and the continuum hypothesis
- 29.
Martin’s axiom is consistent
- 30.
a shorter proof: Martin’s axiom and the continuum hypothesis
- 31.
Zermelo’s postulate
- 32.
Zermelo’s well-ordering theorem
- 33.
Zorn’s lemma
- 34.
example of universe
- 35.
example of universe of finite sets
- 36.
proof of properties of universe
- 37.
Tarski’s axiom
- 38.
universe
- 39.
von Neumann-Bernays-Goedel set theory
- 40.
chain condition
- 41.
composition of forcing notions
- 42.
composition preserves chain condition
- 43.
equivalence of forcing notions
- 44.
forcing
- 45.
forcing relation
- 46.
forcings are equivalent
if one is dense in the other
- 47.
FS iterated forcing preserves chain condition
- 48.
iterated forcing
- 49.
iterated forcing and composition
- 50.
partial order with chain condition does not collapse cardinals
- 51.
proof of partial order with chain condition does not collapse cardinals
- 52.
proof that forcing notions are equivalent to their composition
- 53.
Boolean valued model
- 54.
complete partial orders do not add small subsets
- 55.
proof of complete partial orders do not add small subsets
- 56.
Levy collapse
- 57.
is equivalent to and continuum hypothesis
- 58.
proof of is equivalent to and continuum hypothesis
- 59.
clubsuit
- 60.
diamond
- 61.
combinatorial principle