Fraenkel \(\cal N16(\aleph_{\gamma})\): Levy's Model II | Historical notes
Description: This is an extension of\(\cal N16\) in which \(A\) has cardinality \(\aleph_{\gamma}\) wherecf\((\aleph_{\gamma}) = \aleph_0\); \(\cal G\) is the group of allpermutations on \(A\); and \(S\) is the set of all subsets of \(A\) ofcardinality less that \(\aleph_{\gamma}\)
Parameter(s): This model depends on the following parameter(s): \(\alpha\), \(\alpha\): ordinal where \(\aleph_{\alpha} = \aleph_{\gamma}\) and \(\mbox{cof}(\aleph_{\alpha} = \aleph_0\)
All Forms Known to be True in \(\cal N16(\aleph_{\gamma})\):
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,
183-alpha,
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,
71-alpha,
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 \(\cal N16(\aleph_{\gamma})\):
430-p,
427,
426,
407,
388,
359,
347,
345,
335-n,
334,
333,
286,
264,
262,
261,
260,
259,
258,
257,
256,
255,
239,
218,
202,
193,
192,
188,
181,
174-alpha,
168,
149,
133,
113,
109,
106,
101,
100,
95-F,
87-alpha,
86-alpha,
71-alpha,
67,
66,
44,
43,
40,
39,
28-p,
20,
15,
8,
1,
A minimial list of forms whose truth in this model imply all others that are true in this model: 63-71-91-191
Falses that are implied by others list: 8-15-71-106-133
References for models trues falses list: References Levy [1964], Pincus [1969], Jech [1973b] (p 124),notes 18 and 120(55).
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