Fraenkel \(\cal N9\): Halpern/Howard Model | Back to this models page
Description: \(A\) is a set of atoms with the structureof the set \( \{s : s:\omega\longrightarrow\omega \wedge (\exists n)(\forall j > n)(s_j = 0)\}\)
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 |
|---|---|
| 3 | \(2m = m\): For all infinite cardinals \(m\), \(2m = m\). |
| 6 | \(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable family of denumerable subsets of \({\Bbb R}\) is denumerable. |
| 37 | Lebesgue measure is countably additive. |
| 63 |
\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
|
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 130 | \({\cal P}(\Bbb R)\) is well orderable. |
| 191 | \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). |
| 273 | There is a subset of \({\Bbb R}\) which is not Borel. |
| 305 | There are \(2^{\aleph_0}\) Vitali equivalence classes. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in{\Bbb Q})(x-y=q)\).). \ac{Kanovei} \cite{1991}. |
| 309 | The Banach-Tarski Paradox: There are three finite partitions \(\{P_1,\ldots\), \(P_n\}\), \(\{Q_1,\ldots,Q_r\}\) and \(\{S_1,\ldots,S_n, T_1,\ldots,T_r\}\) of \(B^3 = \{x\in {\Bbb R}^3 : |x| \le 1\}\) such that \(P_i\) is congruent to \(S_i\) for \(1\le i\le n\) and \(Q_i\) is congruent to \(T_i\) for \(1\le i\le r\). |
| 313 | \(\Bbb Z\) (the set of integers under addition) is amenable. (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 361 | In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325. |
| 363 | There are exactly \(2^{\aleph_0}\) Borel sets in \(\Bbb R\). G. Moore [1982], p 325. |
| 368 | The set of all denumerable subsets of \(\Bbb R\) has power \(2^{\aleph_0}\). |
| 369 | If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). |
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 |
|---|---|
| 88 | \(C(\infty ,2)\): Every family of pairs has a choice function. |
| 133 | Every set is either well orderable or has an infinite amorphous subset. |
| 192 | \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). |
| 344 | If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\). |
Historical background: We identify \(A\) with this set to simplifythe description of \(\cal G\). \(\cal G\) is the group of all permutations\(\phi\) on \(A\) such that \(\{a\in A: \phi(a)\ne a\}\) is bounded; and \(S=\{A_0^n: n\in\omega\}\), where \(A_0^n\) is the set of all sequences \(s\)such that \(s_j = 0\) for all \(j\ge n\). (The set \(x\) has support \(A^n_0\) ifall permutations in \(\cal G\) that leave \(A^n_0\) pointwise fixed and leavethe first \(n\) coordinates of each \(s\in A\) pointwise fixed, also leave \(x\)fixed.) In this model, the axiom of choice for 2-element sets (88) isfalse, but the \(2m=m\) principle (3) is true.Form 3 implies that there areno amorphous sets (64) is true and 64 implies that 133 (Every set iseither well orderable or has an infinite amorphous subset.) is false.Since 3 implies 9 (Every Dedekind finite set is finite.) and 70 (There isa non-trivial ultrafilter on \(\omega\).) is true in every FM model, itfollows that 63 (Every infinite set has a non-trivial ultrafilter.) isalso true.
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