Axiomatic set theory

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Axiomatic set theory

• Jouko Väänänen

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Equivalent definition of sum

• ?0 ?
• ?(?1) (??)1
• ? ? lim?lt?(??)

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Equivalent definition of product

• ??0 0
• ??(?1) (???)?
• ??? lim?lt?(???)

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Definition of exponentiation

• ?0 1
• ??1 ????
• ?? lim?lt?(??)

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Continuous functions on ordinals

• fOn?On is continuous if
• f(?)lim?lt? f(?)
• for all limit ordinals ?.
• f(?)?? is continuous
• ? ? lim?lt?(??)
• f(?) ? ? ? is continuous
• ??? lim?lt?(???)
• f(?)?? is continuous
• ?? lim?lt?(??)

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Order-preserving functions on ordinals

• fOn?On is order-preserving if
• ?lt? ?f(?)ltf(?)
• for all ? and ?
• f(?)? ? is order-preserving
• ??? ? ??? ??
• f(?)? ? ? is order-preserving
• ??? ? ??? ? ???
• f(?)?? is order-preserving
• ??? ? ?? ? ??

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Normal functions on ordinals

• fOn?On is normal if it is continuous and
  order-preserving
• f(?)?? is normal
• f(?)? ? ? is normal
• f(?)?? is normal

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Exercise

• If fOn?On is normal, then
• f(lim ?lt? ??) lim?lt? f(??)
• for all increasing sequences (??)?lt?.

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Fixed-point Theorem

• Every normal function has a fixed-point ?f(?).
• ?00
• ?n1f(?n) ??n
• ?limn ?n
• f(?)f(limn ?n) limn f(?n)
• limn ?n1?

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Many fixed-points

• Every normal function has fixed-points
  arbitrarily high
• Given ?
• ?0 ?
• ?n1f(?n)? ?
• ?limn ?n ? ?
• f(?)f(limn ?n) limn f(?n)
• limn ?n1?

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Fixed points of arithmetic functions

• For every ? there is ? such that ???.
• E.g. ????
• For every ? there is ? such that ????.
• E.g. ???
• For every ? there is ? such that ???.
• E.g. there are ? such that ???

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Simultaneous fixed points

• Suppose f and g are normal. Then there is ? such
  that ?f(?) and ?g(?).
• ?00
• ?2n1f(?2n)??2n
• ?2n2g(?2n1) ??2n1
• ?limn ?n
• f(?)f(limn ?2n) limn f(?2n)
• limn ?2n1?
• g(?)g(limn ?2n1) limn g(?2n1)
• limn ?2n2?

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Ordinal substraction

• For all ? and ? such that ?lt? there is a unique ?
  such that ???

?
?
?
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More formal proof

• Let ? be the least ? such that ??gt?.
• Case 1 ? is a limit ordinal. So ???? for all
  ?lt?. But then ??lim?lt? ?? ??lt? ????, that
  is, ????, a contradiction.
• Case 2 ? is a successor ordinal ?1. Since ?lt?,
  ????. But if ??lt?, then ????, a contradiction.
  So ???.
• Uniqueness by the Cancellation Law.

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Properties of exponentiation

• ????????
• (??)?????

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Proof of ????????

• ????0 ???1?? ??0
• ?????1 ??????? ????? ?(??)1??(?1)
• ????? ???lim?lt?(??) lim?lt? ????? lim?lt? ???
• Claim lim?lt? ??? lim?lt?? ?? (???).
• If ?lt?, then ??lt ??, so ???? ???.
• Suppose ?lt??. If ???, then ??????lim?lt? ???.
• Assume then ?lt?, e.g. ? ??. Then ?lt?. Now ??
  ??? ? lim?lt? ???.

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Limit ordinals are divisible by ?

• If ? is a limit ordinal, then ???? for some ?.
• Proof Let ? be the smallest ? such that ?lt???
• Case 1 ? is a limit ordinal. If ?lt?, then ?????.
  Hence ???lim?lt??????, a contradiction.
• Case 2 ? ?1. So ?????. If ???lt?, then ???
  ??(?1) ??????, since ? is a limit ordinal!
  This is a contradiction. Hence in fact ????. QED