CH 2 Ordinals and the Axiom of Choice Flashcards
Construct the ordinals.
State the Principle of Transfinite Induction.
State the Principle of Strong Induction.
Define a well-ordered set.
Why is an ordinal a well-ordered set?
Do the ordinal numbers form a set?
State a proposition comparing ‘less than’ to membership and subset inclusion.
Construct an order-preserving bijection between well-ordered sets A and B.
State and prove a proposition on ordinals and well-ordered sets which this construction implies.
Define a cardinal number.
Is omega (w) a cardinal number?
State the Well-Ordering Principle.
State the Axiom of Choice.
Reformulate it in terms of choice functions.
State and prove (by construction) a theorem on families of 2-element sets.
Zorn’s Lemma: partially ordered set, chain, upper bound, maximal element
Recall a well-ordering and the Well-Ordering Principle.
State and prove the implication between WOP and AoC.
State Zorn’s Lemma.
Also, every vector sapace has a basis by Zorn’s Lemma.
Prove Zorn’s Lemma implies the Well-Ordering Principle.
Prove that the Axiom of Choice implies Zorn’s Lemma.
Also, assuming the Axiom of Choice, then there exists a non-measurable set of real numbers.
