Fraenkel \(\cal N12(\aleph_{\alpha})\): A generalization of \(\cal N12(\aleph_1)\).Replace ``\(\aleph_1\)'' by ``\(\aleph_{\alpha}\)'' where \(\aleph_{\alpha}\) isa singular cardinal | Historical notes
Description: Thus, \(|A|=\aleph_{\alpha}\); \(\cal G\) is the groupof all permutations on \(A\); and \(S\) is the set of all subsets of \(A\) withcardinality less than \(\aleph_{\alpha}\)
Parameter(s): This model depends on the following parameter(s): \(\alpha\), \(\alpha\): \(\aleph_{\alpha}\) a singular cardinal, \(\alpha > 0\)
All Forms Known to be True in \(\cal N12(\aleph_{\alpha})\):
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,
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,
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,
43,
42,
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 \(\cal N12(\aleph_{\alpha})\):
193,
192,
163,
133,
15,
3,
2,
1,
A minimial list of forms whose truth in this model imply all others that are true in this model: 63-87-91-191
Falses that are implied by others list: 3-15-133-163-192-193
References for models trues falses list: References Jech [1973b] (p 124, Thm 8.6), Monro [1974], Blass [1979], Brunner [1982a], notes 18, 60, 104, and 120(4).
Back