Models

Cohen \(\cal M17\): Gitik's Model | Back to this models page

Description: Using the assumption that for every ordinal \(\alpha\) there is a strongly compact cardinal \(\kappa\) such that \(\kappa >\alpha\), Gitik extends the universe \(V\) by a filter \(G\) generic over a proper class of forcing conditions

When the book was first being written, only the following form classes were known to be true in this model:

Form Howard-Rubin Number Statement
0  \(0 = 0\).

When the book was first being written, only the following form classes were known to be false in this model:

Form Howard-Rubin Number Statement
91

\(PW\):  The power set of a well ordered set can be well ordered.

104

There is a regular uncountable aleph. Jech [1966b], p 165 prob 11.26.

182

There is an aleph whose cofinality is greater than \(\aleph_{0}\).

315

\(\Omega = \omega_1\), where
\(\Omega = \{\alpha\in\hbox{ On}: (\forall\beta\le\alpha)(\beta=0 \vee (\exists\gamma)(\beta=\gamma+1) \vee\)
there is a sequence \(\langle\gamma_n: n\in\omega\rangle\) such that for each \(n\),
\(\gamma_n<\beta\hbox{ and } \beta=\bigcup_{n<\omega}\gamma_n.)\} \)

Historical background: (Inaccessible cardinals are used inthe construction.) In \(V[G]\), he constructs a symmetric submodel \(\calM17\) in which every infinite set is a countable union of sets of smallercardinality. Thus, there is no regular uncountable cardinal (104 isfalse), there is no aleph whose cofinality is greater than \(\aleph_0\) (182is false), and \(\Omega =\) On (315 is false). See Note 55.

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