Axiomatic set theory

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Axiomatic set theory

• Jouko Väänänen

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Notation

• Sets capital letters X,Y,...
• Elements small letters x,y,

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Partial order (X,?), poset

• Reflexive, antisymmetric and transitive
• Example (P(X), ?)

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Lemma

• Suppose (X, ?) is a poset. Then there is a set Y
  such that (X, ?) (Y,?).
• Proof (Lecture)

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Well-founded poset

• Minimal element none smaller
• Partial order is well-founded if every non-empty
  subset has a minimal element.

Minimal element
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Lemma

• A poset is well-founded iff it has no infinite
  descending sequence.
• Proof (lecture)

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Induction on a well-founded set

• Suppose (X,?) is a well-founded poset.
• Suppose E?X is such that
• If x?X and every y?X with yltx is in E, then x?E.
  
• Then EX.

x
X
E
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Proof

• Suppose X-E??.
• Let x be lt-minimal element in X-E.
• Then every yltx is in E.
• Hence by assumption x?E, a contradiction.
• So EX. QED

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Total (linear) ordering

• Poset that is connected i.e. if a?b, then a ? b
  or b ? a.

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Wellordering, wellordered set, woset

• Linear ordering that is well-founded.

...
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Total (linear) ordering

• Poset that is connected i.e. if a?b, then a ? b
  or b ? a.

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Wellordering, wellordered set, woset

• Linear ordering that is well-founded.

...
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Order isomorphism

• (X,?), (X, ?) wosets
• f X?X is an order isomorphism if
• f is a bijection
• xlty ? f(x) lt f(y)
• Write fX X

X
f
X
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Important Theorem

• Let (X,?) be a woset and Y?X.
• If fX Y, then for all x x ? f(x)

Y
f
X
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Proof
x1 f(x0)
f(x1)
Y
f
X
x0
x1
Let E x f(x)ltx
Consider x1 f(x0).
Suppose x0 min(E).
f(x1)ltf(x0)x1
x1ltx0
x1? E
Contradiction, as x0 was minimal!