Fraenkel \(\cal N11\): Jech's Model II | Back to this models page

Description: Let \((I,\precsim)\) be a partially ordered set inthe kernel (in the base model without atoms)

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
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.

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
15

\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every  set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)).  

59-le

If \((A,\le)\) is a partial ordering that is not a well ordering, then there is no set \(B\) such that \((B,\le)\) (the usual injective cardinal ordering on \(B\)) is isomorphic to \((A,\le)\).
Mathias [1979], p 120.

192

\(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\).

Historical background: Let \(A=\{a_{i,n}: i\in I,n\in\omega\}\). For each \(p\subseteq I\), let \(S_p=\{a_{i,n}: i\in p,n\in\omega\}\). \(\cal G\) is the group of all permutations of \(A\) that leave\(S_{\{i\}}\) fixed, for each \(i\in I\). \(S\) is the set of all finite subsetsof \(A\). In this model, there is a set of cardinals that is isomorphic to\((I,\precsim)\). Thus,Form 59(\(\precsim\)) is false.

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