2 Ordinals and Axiom of Choice Flashcards
1
Q
Construct the ordinals.
A
2
Q
State the Principle of Transfinite Induction and Principle of Strong Induction.
A
3
Q
Define a well-ordered set.
Define an ordinal.
How is the Burali-Forti paradox relevant here?
A
4
Q
State a ‘useful property’ of ordinals (a 2-part iff statement).
State a useful proposition on bijections and ordinal numbers.
A
5
Q
Define a cardinal number.
State the Well-Ordering Principle.
A
6
Q
State the Axiom of Choice.
Reformulate the AC using choice functions.
State the theorem on families of 2-element sets.
A
7
Q
Prove that the Well-ordering Principle implies the Axiom of Choice.
A
8
Q
First define a chain in P, upper bound on C, and maximal element in P.
State Zorn’s Lemma
A
9
Q
Prove Zorn’s Lemma implies the Well-ordering Principle.
Prove the Axiom of Choice implies Zorn’s Lemma.
A
