Axiom Of Choice

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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