Fraenkel \(\cal N10\): Höft/Howard/Mostowski Model | Historical notes
Description: (The model is a variation of\(\cal N3\).) \(A\) as ordered by \(\precsim\) has the same order type as therationals; \(\cal G\) is the group of all order automorphisms of \(A\); \(S\) isthe set of all subsets \(E\) of \(A\) that satisfy the following threeconditions:\item{1.} \(E\) is well ordered by \(\precsim\).\item{2.} \(E\) is bounded in \(A\).\item{3.} If \(b:\alpha\to E\) is an order preserving bijection from\(\alpha\) onto \(E\) and if \(\lambda < \alpha\) is a limit ordinal then\(\{b(\beta) : \beta < \gamma\}\) has no least upper bound in\((A,\precsim)\).\par\noindentIn H\"oft/Howard [1994] it is shown that, in \(\cal N10\), everyDedekind finite set is finite (9 is true), but \((A,\precsim)\) is alinearly ordered set with no infinite descending sequences that cannot bewell ordered (77 is false)
Parameter(s): This model does not depend on parameters
All Forms Known to be True in \(\cal N10\):
423,
404,
390,
389,
387,
378,
377,
376,
374-n,
373-n,
371,
369,
368,
367,
366,
364,
363,
362,
361,
358,
342-n,
336-n,
325,
313,
309,
307,
306,
305,
304,
296,
289,
288-n,
280,
273,
272,
252,
251,
249,
223,
222,
221,
217,
216,
212,
211,
206,
203,
199(\(n\)),
198,
197,
194,
191,
190,
189,
185,
182,
170,
169,
167,
145,
142,
139,
137-k,
132,
130,
128,
127,
124,
108,
104,
98,
94,
93,
92,
91,
84,
83,
82,
80,
79,
74,
73,
70,
64,
63,
57,
38,
37,
35,
34,
19,
18,
17,
13,
12,
11,
10,
9,
6,
5,
0,
All Forms Known to be False in \(\cal N10\):
430-p,
427,
409,
408,
407,
399,
391,
388,
384,
359,
347,
345,
344,
343,
335-n,
334,
333,
332,
331,
326,
323,
317,
316,
303,
295,
292,
286,
284,
270,
264,
262,
261,
260,
259,
258,
257,
256,
255,
239,
218,
213,
202,
201,
193,
192,
188,
174-alpha,
168,
149,
133,
123,
114,
112,
109,
107,
106,
101,
100,
95-F,
90,
89,
88,
87-alpha,
85,
77,
68,
67,
66,
62,
61,
60,
51,
50,
49,
44,
43,
40,
36,
30,
28-p,
20,
15,
14,
1,
A minimial list of forms whose truth in this model imply all others that are true in this model: 9-63-91-191
Falses that are implied by others list: 77-88-106
References for models trues falses list: ReferencesH\"oft/Howard [1994], notes 18, 113, and 120(55 and 56).
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