Axiom Of Choice

Cohen $\cal M36$ : Figura's Model | Back to this models page

Description: Starting with a countable, standard model, $\cal M$ , of $ZFC + 2^{\aleph_0}=\aleph_{\omega +1}$ , Figura uses forcing conditions that are functions from a subset of $\omega\times\omega$ to $\omega_\omega$ to construct a symmetric extension of $\cal M$ in which there is an uncountable well ordered subset of the reals (Form 170 is true), but $\aleph_1= \aleph_{\omega}$ so $\aleph_1$ is singular (Form 34 is false)

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
170	$\aleph_{1}\le 2^{\aleph_{0}}$ .

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
6	$UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)$ : The union of a denumerable family of denumerable subsets of ${\Bbb R}$ is denumerable.
34	$\aleph_{1}$ is regular.
91	$PW$ : The power set of a well ordered set can be well ordered.

Historical background: (In fact, Figura proves this result for any twocardinals $\kappa$ (replacing $\aleph_0$ ) and $\mu$ (replacing $\aleph_\omega$ ) such that cf $(\kappa)=\kappa\le\hbox{cf}(\mu)<\mu$ and $2^\kappa= \mu^+$ in $\cal M$ . See Note 3.) It is showm inHoward/Keremedis/Rubin/Stanley/Tachtsis [1999] that 170 + 6( $UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)$ implies 34. Consequently,Form 6is also false in $\cal M36$ .

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