Fraenkel \(\cal N2(\aleph_{\alpha})\): Jech's Model | Historical notes
Description: This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\)
Parameter(s): This model depends on the following parameter(s): \(\alpha\), \(\alpha\): ordinal number
All Forms Known to be True in \(\cal N2(\aleph_{\alpha})\):
423,
420,
419,
418,
412,
411,
404,
390,
389,
388,
387,
380,
379,
378,
377,
376,
375,
374-n,
373-n,
371,
369,
368,
367,
366,
365,
364,
363,
362,
361,
358,
357,
355,
354,
353,
352,
351,
350,
342-n,
341,
340,
339,
338,
336-n,
325,
313,
309,
307,
306,
305,
304,
296,
289,
288-n,
282,
280,
279,
273,
272,
252,
251,
249,
243,
223,
222,
221,
217,
216,
212,
211,
209,
206,
203,
199(\(n\)),
198,
197,
196-alpha,
194,
191,
190,
189,
185,
183-alpha,
182,
174-alpha,
170,
169,
167,
155,
154,
150,
145,
142,
139,
137-k,
135,
132,
131,
130,
128,
127,
126,
124,
119,
113,
108,
106,
104,
98,
94,
93,
92,
91,
87-alpha,
86-alpha,
84,
83,
82,
80,
79,
78,
77,
74,
73,
71-alpha,
70,
64,
63,
57,
44,
43,
42,
41,
39,
38,
37,
35,
34,
32,
31,
29,
27,
26,
24,
19,
18,
17,
16,
13,
12,
11,
10,
9,
8,
6,
5,
0,
All Forms Known to be False in \(\cal N2(\aleph_{\alpha})\):
422-n,
409,
408,
407,
403,
402,
401,
400,
399,
398,
394,
393,
392,
391,
384,
359,
347,
345,
344,
343,
335-n,
333,
332,
331,
327,
326,
324,
323,
322,
317,
303,
302,
295,
286,
284,
270,
264,
262,
261,
260,
259,
258,
257,
256,
255,
250,
239,
231,
218,
213,
202,
201,
193,
192,
168,
165,
163,
151,
133,
123,
122,
121,
111,
110,
109,
107,
101,
100,
95-F,
88,
85,
68,
66,
62,
61,
60,
50,
49,
45-n,
40,
36,
30,
23,
21,
20,
15,
14,
3,
2,
1,
A minimial list of forms whose truth in this model imply all others that are true in this model: 44-63-87-91-135-191
Falses that are implied by others list: 3-45-88-111-163-192-193
References for models trues falses list: References Blass [1979], Brunner [1982a], Jech [1966b], \cite{1973b}, Mostowski [1948], van Douwen [1980], notes 18, 60, 104, and 120(56).
Back