Axiomatic set theory

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Axiomatic set theory

• Jouko Väänänen

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Hausdorffs Maximality Principle HP
Every chain can be extended to a maximal chain
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ZL implies HP

• Given a poset P.
• The set of all chains in P forms a poset (P,?)

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Equivalents of the Axiom of Choice
Axiom of Choice
Axiom of Choice
Well-Ordering Principle
Today
Zorns Lemma
Hausdorffs Principle
Tukeys Lemma
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Finite character

• A set X has finite character if
• an arbitrary set Y is in X if and only if every
  finite subset of Y is in X
• Examples
• Fix A and B. The set of all f?AxB that are
  functions.
• Fix A and B. The set of all f?AxB that are
  one-one functions.
• Fix A. The set of all f?Ax(?A) that are functions
  and satisfy f(y)?y for all y?A.

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Tukeys Lemma TL

• For every set X and every non-empty Y?P(X) with
  finite character there is a ?-maximal element in
  Y.

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TL implies AC

• Suppose A is a set of non-empty sets.
• Let X be the set of functions f such that dom(f)
  ?A and f(a)?a for all a?dom(f).
• X has finite character.
• Let f be a maximal element.
• If dom(f)?A, let a?A-dom(f). Let b?a.
• gf?(a,b)?X, a contradiction.
• So f is as required by AC. QED

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HP implies TL

• Given set X and a set Y of subsets of X with
  finite character.
• By HP there is a maximal ?-chain H of elements of
  Y.
• Let A ?H. We show this is the maximal element of
  Y needed for TL.
• If H has a largest element, it has to be A and
  then A is in Y and we are done.
• If H has no largest element, then A is in Y by
  finite character. So in fact A has to be in H, a
  contradiction.