Fraenkel \(\cal N14\): Morris/Jech Model | Back to this models page

Description: \(A = \bigcup\{A_{\alpha}: \alpha <\omega_1\}\), where the \(A_{\alpha}\)'s are pairwise disjoint, each iscountably infinite, and each is ordered like the rationals; \(\cal G\) isthe group of all permutations on \(A\) that leave each \(A_{\alpha}\) fixedand preserve the ordering on each \(A_{\alpha}\); and \(S = \{B_{\gamma}:\gamma < \omega_1\}\), where \(B_{\gamma}= \bigcup\{A_{\alpha}: \alpha <\gamma\}\)

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.

43

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

63

\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
Jech [1973b], p 172 prob 8.5.

91

\(PW\):  The power set of a well ordered set can be well ordered.

130

\({\cal P}(\Bbb R)\) is well orderable.

144

Every set is almost 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(\(n\)).  

103

If \((P,<)\) is a linear ordering and \(|P| > \aleph_{1}\) then some initial segment of \(P\) is uncountable. Jech [1973b], p 164 prob 11.21.

133  

Every set is either well orderable or has an infinite amorphous subset.

163

Every non-well-orderable set has an infinite, Dedekind finite subset.

192

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

Historical background: Let \(P= A\cup\omega_1\) and order \(P\) so that the elements in\(A_{\alpha}\) precede those in \(A_{\beta}\) if \(\alpha < \beta\), theelements in each \(A_{\alpha}\) are ordered like the rationals as above, andeach \(\alpha\in\omega_1\) precedes \(A_\alpha\) and follows all \(A_\beta\) for\(\beta <\alpha\). Then \(P\) is a linearly ordered set such that\(|P|>\aleph_1\), but each initial segment of \(P\) is countable. Thus, form103 is false. In Note 144 we show dependent choice (form 43) is true andin Note 155 we show that ``every set is almost well orderable'' (form 144)is true.

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