Axiom Of Choice

Cohen $\cal M2$ : Feferman's model | Back to this models page

Description: Add a denumerable number of generic reals to the base model, but do not collect them

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
40	$C(WO,\infty)$ : Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
142	$\neg PB$ : There is a set of reals without the property of Baire. Jech [1973b], p. 7.
165	$C(WO,WO)$ : Every well ordered family of non-empty, well orderable sets has a choice function.
204	For every infinite $X$ , there is a function from $X$ onto $2X$ .

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
70	There is a non-trivial ultrafilter on $\omega$ . Jech [1973b], prob 5.24.
88	$C(\infty ,2)$ : Every family of pairs has a choice function.
152	$D_{\aleph_{0}}$ : Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for $D_{\kappa}$ , $\kappa$ a well ordered cardinal.)
163	Every non-well-orderable set has an infinite, Dedekind finite subset.
203	$C$ (disjoint, $\subseteq\Bbb R)$ : Every partition of ${\cal P}(\omega)$ into non-empty subsets has a choice function.
222	There is a non-principal measure on $\cal P(\omega)$ .
274	There is a cardinal number $x$ and an $n\in\omega$ such that $\neg(x$ adj $^n\, x^2)$ . (The expression `` $x$ adj $^n\, ya$ " means there are cardinals $z_0,\ldots, z_n$ such that $z_0 = x$ and $z_n = y$ and for all $i,\ 0\le i < n,\ z_i< z_{i+1}$ and if $z_i < z\le z_{i+1}$ , then $z = z_{i+1}.)$ (Compare with [0 A]).
385	Countable Ultrafilter Theorem: Every proper filter with a countable base over a set $S$ (in ${\cal P}(S)$ ) can be extended to an ultrafilter.

Historical background: Feferman has shown that in this model there is an infinite set of pairs with no choice function (88 is false), there is a partition of $\cal P(\omega)$ with no choice function(203 is false) and the countable ultrafilter theorem (385) is false.Truss [1978] has shown that in this model $C(WO,\infty)$ (40)is true. (He gives credit to Solovay for the proof.) In Truss [1978] it is also shown that $\cal P(\omega)$ does not have anon-trivial ultrafilter (70 is false), and for every infinite set $x$ there is a mapping of $x$ onto $2\times x$ (204) is true. (See Note 65.)Pincus shows that there is a set of reals that does not have the propertyof Baire (142 is true), but $\cal P(\omega)$ has no nontrivial real valuedmeasure which is 0 on finite sets (222 is false). (The proof ofForm 142requires that the Continuum Hypothesis holds in the outer model.) Since40 implies 8 ( $C(\aleph_0,\infty)$ ), it follows from Brunner [1982a] that in this model there is a set that cannot be well orderedand does not have an infinite Dedekind finite subset, (163 is false).(Form 8 plusForm 163 iff AC.) Truss [1974b] proves that thereis a sequence $x_1, x_2, \cdots$ of cardinals such that for each $n$ , $x_n=\aleph_{n+1} + 2^{\aleph_0}$ adj $\aleph_{n+1} + \aleph_2\cdot\2^{\aleph_0}$ adj $\aleph_{n+1} + \aleph_3\cdot\ 2^{\aleph_0}$ adj $\cdots$ adj $\aleph_{n+1} + \aleph_{n+1}\cdot 2^{\aleph_0} = x^2_n$ . (Thesymbol $x$ adj $y$ means $x < y$ and there is nothing between $x$ and $y$ .) It follows that 274 is false.

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