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 \(\cal M36\).
Back