Fraenkel \(\cal N4\): The Mathias/Pincus Model I | Back to this models page
Description: \(A\) is countably infinite;\(\precsim\) is a universal homogeneous partial ordering on \(A\) (SeeJech [1973b] p 101 for definitions.); \(\cal G\) is the group ofall order automorphisms on \((A,\precsim)\); and \(S\) is the set of allfinite subsets of \(A\)
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. |
| 64 | \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) |
| 90 | \(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133. |
| 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. |
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 |
|---|---|
| 30 | Ordering Principle: Every set can be linearly ordered. |
| 83 | \(E(I,II)\) Howard/Yorke [1989]: \(T\)-finite is equivalent to finite. |
| 89 | Antichain Principle: Every partially ordered set has a maximal antichain. Jech [1973b], p 133. |
| 114 | Every A-bounded \(T_2\) topological space is weakly Loeb. (\(A\)-bounded means amorphous subsets are relatively compact. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.) |
| 133 | Every set is either well orderable or has an infinite amorphous subset. |
Historical background: In \(\cal N4\), every linearly ordered set can bewell ordered (90) is true but the Antichain Principle (89) is false. Thereare no amorphous sets (64 is true) and the set of atoms is an infinite \(T\)finite set (83 is false). (See Note 105.) SinceForm 64 is true and AC isfalse,Form 133 (Every set is either well orderable or has an infiniteamorphous subset.) is false. The set of atoms with the discrete topologyis a \(T_2\)-space which is not weakly Loeb (114 is false). SinceForm 118(Every orderable topological space is normal.) is true (90 implies 118),it follows from Krom [1986] that not every set can be linearlyordered (30 is false). (The Kinna-Wagner Principle (15) is false in everyFM model and Krom proves that \(\neg 15 + 30 \to\neg 118\).)
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