Notes

Description: Hickman [1980b] generalizes the notion of amorphous.

Content:

Hickman [1980b] generalizes the notion of amorphous as follows:

Definition:

  1. \(X\) is an \(\aleph_{\alpha}\)set if \(|X|\) is incomparable with \(\aleph_{\alpha}\)  and for all \(\beta  < \alpha\), \(\aleph_{\beta} < |X|\).
  2. \(X\) is \(\aleph_{\alpha}\) minimal if \(X\) is an \(\aleph_{\alpha}\) set and for all \(Y\subseteq X\), either \(|Y| <\aleph_{\alpha}\) or \(|X-Y|<\aleph_{\alpha}\), (abbreviated \(X\) is \(\alpha\)-m).
  3. \(X\) is weakly \(\aleph_{\alpha}\) minimal if \(X\) is an \(\aleph_{\alpha}\) set and for all \(Y\subseteq X\) at most one of \(Y\) and \(X - Y\) is an \(\aleph_{\alpha}\) set, (\(X\) is \(\alpha\)-wm).
  4. \(X\) is pseudo \(\aleph_{\alpha}\) minimal if \(X\) is an\(\aleph_{\alpha }\) set and for all \(Y\subseteq X\), exactly one of \(Y\) and \(Y - X\) is an \(\aleph _{\alpha }\) set, (\(X\) is \(\alpha\)-pm).

The following are proved:

Theorem 1: \(X\) is \(0\)-m iff \(X\) is \(0\)-wm iff \(X\) is 0-pm.

Theorem 2:

  1. For all \(\alpha\), \(X\) is \((\alpha +1)\)-pm implies \(X\) is \((\alpha +1)\)-wm and
  2. \(X\) is \((\alpha +1)\)-wm iff \(X\) is \((\alpha +1)\)-m.

Theorem 3: If \(\alpha\) is a limit ordinal \((\alpha\neq  0)\), then

  1. \(X\) is \(\alpha\)-m implies \(X\) is \(\alpha \)-wm.
  2. \(X\) is \(\alpha\)-wm iff \(X\) is \(\alpha \)-pm.

Theorem 4:  For all ordinals \(\alpha \), \(Con(ZF + \exists X (X\) is \(\alpha\)-m\())\).

Theorem 5:

  1. For all ordinals \(\alpha\), the following is not a theorem of \(ZF\): \[\forall X ( X\hbox{ is }(\alpha +1)\hbox{-m}\rightarrow  X\hbox{ is }(\alpha +1)\hbox{-mp})\]
  2. For all limit ordinals \(\alpha\), it is not a theorem of \(ZF\) that \[\forall X (X\hbox{ is }\alpha\hbox{-mp}\rightarrow X\hbox{ is }\alpha\hbox{-m}).\]

Theorem 6: If \(\aleph_{\alpha}\) is regular, then \(Con(ZF + \) there is a linearly ordered \((\alpha +1)\)-pm set\()\).

Theorem 7: For any \(\alpha\), \(Con(ZF + \) there is a linearly ordered \((\alpha +1)\)-m set\()\).

Theorem 8: If \(\alpha\) is a limit ordinal there are no linearly ordered \(\alpha\)-wm sets.

Open Problem: The consistency of the existence of linearly ordered \((\alpha +1)\)-pm set where \(\aleph_{\alpha}\) is not regular.

Howard-Rubin number: 56

Type: Definitions and summaries

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