Notes

Description: In Truss [1975] weakenings of König's lemma are considered

Content:

In Truss [1975] the following weakenings of König's lemma are considered:

For any subset \(Z\subseteq\omega\), \(KL(Z)\) is the formula "If \(T\) is an infinite tree such that for every element \(a\) of \(T\) ,the number of immediate successors of \(a\) is in \(Z\), then \(T\) has an infinite branch." Form 249 is \(KL(\{2\})\). Here is a summary of the results:

  1. Form 250 (\((\forall  n \in  \omega )(C(WO,n)) \rightarrow KL(\{k\})\) is not a theorem of  \(ZF^{0}\) for any \(k > 1.\)
  2. Form 80 (\(C(\aleph _{0},2)) \rightarrow  KL(\{2\})\) is not a theorem of \(ZF^{0}\).
  3. The following conditions are equivalent:
    1. \(ZF^0 \vdash KL(Z) \rightarrow  KL(\{n\})\)
    2. \(ZF^0 \vdash KL(Z) \rightarrow  C(\aleph _{0},n)\)
    3. \(ZF^0 \vdash (\forall  m \in  Z)( KL(\{m\})) )\rightarrow  C(\aleph _{0},n)\)
    4. Gauntt's condition \(L(n,Z)\), see Note 15.

Howard-Rubin number: 87

Type: Reference summary

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