Models

Fraenkel \(\cal N60\): de la Cruz-Hall model 3 | Historical notes

Description: Let \(\{ R_{n,i} : n, i \in \omega \}\) be a partition of \(A\) (the set of atoms) into continuum sized sets and fix bijections \(f_{n,i} : \mathbb R \to R_{n.i}\). For each \(n \in \omega\) let \(A_{n} = \bigcup \{ R_{n,i} : i \in \omega \}\) and let \(D_{n}\) be the metric on \(A_{n}\) such that each \(f_{n,i}\) is an isometry and the distance betweenn elements in different \(R_{n,i}\)'s is always \(1\). Let \(G^{+}\) be the group of permutations \(\pi\) of A such that

  1. \(\pi\) is the identity on all but finitely many of the \(R_{n,i}\)'s.
  2. For all \(n\) and \(i\) in \(\omega\), there is a \(j \in \omega\) such that \(\pi(R_{n,i}) = R_{n,j}\) (so that \(\pi(A_{n}) = A_{n}\)).
  3. If \(\pi(R_{n,i}) = R_{n,j}\) then \(f^{-1}_{n,j} \circ \pi \circ f_{n,i}\) is an affine transformation of \(\mathbb R\).
Let \(\mathcal D = \{ \pi(D_{n}) : n \in \omega \mbox{ and } \pi \in G^{+} \}\). The group \(G\) used to define the permutation model is the group generated by elements \(g \in G^{+}\) such that \(g\) is an isometry of some \(D \in \mathcal{D}\). Let \(\mathcal{W}\) be the set of relations \(w\) such that \(w\) is a well-ordering of some \(R_{n,i}\). The set of supports is the set of finite subsets of \(\mathcal{D} \cup \mathcal{W}\).

Parameter(s): This model does not depend on parameters

All Forms Known to be True in \(\cal N60\):
320, 191, 91, 31,

All Forms Known to be False in \(\cal N60\):
192, 15, 418,

A minimial list of forms whose truth in this model imply all others that are true in this model:

Falses that are implied by others list:

References for models trues falses list: De la Cruz, Howard, Keremedis, Rubin [2002A], Note 118

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