ZFC
CHOICE
ORDERS
ELEMENTARY ZFC
SEP
Erik A. Andrejko
University of Wisconsin - Madison
Summer 2007
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
ZERMELO FRAENKEL CHOICE
FIGURE: Ernst Zermelo
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
LANGUAGE
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
ZFC
EXTENSIONALLY
∀z[z ∈ x ⇐⇒ z ∈ y ] =⇒ x = y
That is, a set of is uniquely determine by its members.
FOUNDATION
∀x[∃y(y ∈ x) =⇒ ∃y[y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y )]]
from this we can conclude that ∀x[x ∈ x].
/
COMPREHENSION For each formula ϕ without y free
∃y∀x[x ∈ y ⇐⇒ x ∈ z ∧ ϕ]
Note that ϕ may mention any parameters except y .
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
ZFC
PAIRING
∃z[x ∈ z ∧ y ∈ z]
That is, we may form a set {x, y } for any sets x and y .
UNION
∀F∃A∀Y ∀x[x ∈ Y ∧ Y ∈ F =⇒ x ∈ A]
That is, for any family F (set) of sets we may form a set
A ⊃ F that contains the union of the family. Using
comprehension we may form the actual set F.
REPLACEMENT For each formula ϕ without y free
∀x ∈ A∃!y ϕ(x, y ) =⇒ ∃Y ∀x ∈ A∃y ∈ Y ϕ(x, y )
If we consider ϕ(x, ) to be a function f which assigns
y = f (x), then the replacement scheme says that the
range of the function is a set.
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
ZFC
INFINITY We define S(x) to be the successor function. That is
S(x) = x ∪ {x}. Then
∃x[0 ∈ x ∧ ∀y ∈ x[S(y ) ∈ x]]
That is, there exists a set x such that x is infinite.
POWER SET
∀x∃y∀z[z ⊆ x =⇒ z ∈ y ]
That is, for any set, the powerset (set of all subsets) of x
exists.
CHOICE
∀A∃R[R well orders A]
where R is an total order on A. That is, every set may
be well ordered.
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
ZF
DEFINITION
ZF All axioms except Choice.
ZFC All axioms including Choice.
DEFINITION
ZF − ZF except foundation.
Con(ZF −) =⇒ Con(ZF )
WARNING
Choice is controversial.
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
CHOICE
THEOREM
T.F.A.E
Every set has a well order.
1
Every family of nonempty sets has a choice function.
2
Zorn’s Lemma: If every chain in a partial order P has an upper
3
bound in P then P has a maximal element.
Tukey’s Lemma: A set X is say to have finite character if
4
Y ∈ X ⇐⇒ ∀k ∈ Y <ω [k ∈ X ]
For X = 0 with finite character then X has a ⊆-maximal element.
/
Basis Theorem: Every vector space has a basis.
5
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
CHOICE IS COMPLICATED
Relative Strength of Various Choice Principals
AC
KS
Well

Basis
+3 Zorn’s ks +3 Tukey’s ks
p2 = p ks +3 Ordering ks +3
Theorem
Lemma Lemma
w Principal
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www
w
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Selection
+3 Prime Ideal Dependant
Compactness ks
Principal
Theorem Choice
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Order Extension
+3 Ordering ACℵ0
+3 ACF ks
r 4<
Principal Principal
rrr
rrr
r
rrr
r
rr
rrr 
r
r
AC2
ACW

TAC ks +3 LEM
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
SCHRÖDER-BERNSTEIN
THEOREM (SCHRÖDER-BERNSTEIN)
B and B A then A ≈ B.
If A
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
ORDERS
DEFINITION
(P, ≤) is a partial order if ≤ is transitive, anti-symmetric and reflexive.
DEFINITION
1 If p ≤ q or q ≤ p then p and q are comparable.
If there is some r such that r ≤ p and r ≤ q then p and q are
2
compatible. Denoted p ⊥ q.
Otherwise p and q are incomparable. Denoted p ⊥ q.
3
A ⊆ P is an anti-chain if for all p, q ∈ A, p ⊥ q.
4
C ⊆ P is a chain if for all p, q ∈ A, p and q are comparable.
5
DEFINITION
(P, ≤) is a linear order if P is a chain.
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
ORDER TYPES
DEFINITION
If (P, ≤) is an order then P ∗ = (P, ≥).
Order types:
ω and ω ∗ ,
1
Q, a dense linear order,
2
3 R
DEFINITION
(A, R) is well founded if every B ⊆ A has an R least element.
ERIK A. ANDREJKO ELEMENTARY ZFC

ZFC
CHOICE
ORDERS
TRANSFINITE INDUCTION
∀α[ϕ(α)]
1 ϕ(0),
ϕ(α) =⇒ ϕ(α + 1),
2
For limit β ,
3
∀ξ < β [ϕ(ξ )] =⇒ ϕ(β )
ERIK A. ANDREJKO ELEMENTARY ZFC