Notes

Description: Definitions for Form 282 and Form [0 AD]

Content:

Definitions for Form 282 and Form [0 AD] and Kleinberg/Seiferas [1973]

Definition: Assume \(\kappa\ge\omega\) and \(\gamma\ge 1\) are well ordered cardinals and \(\alpha\) and \(\beta\) are ordinals. Then \(\kappa \to (\beta)^{\alpha}_{\gamma}\) denotes the partition \iput{\(\kappa \to (\beta)^{\alpha}_{\gamma}\)} relation ``For every function \(F\) mapping the set of \(\alpha\)-sequences from \(\kappa\) into \(\gamma\) there exists a subset \(X\) of \(\kappa\) of order type \(\beta\) such that the range of \(F\) on the set of \(\alpha\) sequences from \(X\) has cardinality 1.''  If \(\gamma = 2\) we write \(\kappa \to (\beta)^{\alpha}\).  (An \(\alpha\)-sequence from \(Y \subseteq On\) is an order preserving map from \(\alpha\) to \(Y\).) \(\kappa \not\to (\beta)^{\alpha}_{\gamma}\) denotes the negation of \(\kappa \to (\beta)^{\alpha}_{\gamma}\).

Form [0 AD] is:  For any finite \(n\), for all ordinals \(\beta\) and well ordered cardinals \(\gamma\), there is a well ordered cardinal \(\kappa\) such that \(\kappa \to (\beta)^{n}_{\gamma}\) and Form 282 is \(\omega \not \to (\omega)^{\omega}\). The axiom of choice implies for any \(\alpha \ge \omega\), given any well ordered cardinal \(\gamma > 1\) and ordinal \(\beta\) there is no \(\kappa \ge \omega\) for which \(\kappa \to (\beta)^{\alpha}_{\gamma}\).  Also proved in Kleinberg/Seiferas [1973] are the following:

Theorem: Assume \(\gamma > 1\) is a well ordered cardinals and \(\alpha\) and \(\beta\) are ordinals with \(\alpha \le \beta\) then

  1. \(C(WO,\infty)\) (which is Form 40) implies \(\neg \exists \kappa (\kappa \to (\beta)^{\alpha}_{\gamma})\) provided
    (*) \(\alpha \ge \omega + \omega\) ordinal addition or \(\beta \ge \lfloor \alpha \rfloor +\omega + \lceil \alpha \rceil \) or \(\gamma \ge \omega\)
  2. If (\(*\)) fails then \(\exists \kappa(\kappa \to (\beta)^{\alpha}_{\gamma}) \Leftrightarrow \omega \to (\omega)^{\omega}\)
  3. Assume \(C(WO,\infty)\) and ``\({\Bbb R}\) can be well ordered'' then for all \(\alpha ,\ \beta\) and \(\gamma\), \(\neg \exists \kappa (\kappa \to (\beta)^{\alpha}_{\gamma})\)
  4. \(C(\aleph_{\alpha},\infty)\) (which is Form 86) implies \(\aleph_{\alpha}\not \to (\omega + \omega)^{\omega + \omega}\).
  5. \(C(\aleph_{\alpha},\infty) \Rightarrow \aleph_{\alpha}\not \to (\omega + \omega)^{\omega}\).
  6. \(C(\aleph_\alpha,\infty) \Rightarrow \aleph_{\alpha}\not \to (\omega)^{\omega}_{\omega}\).  (So Form 8 (\(C(\aleph_0,\infty)\)) implies Form 282.

Howard-Rubin number: 97

Type: Definitions

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