Fraenkel \(\cal N12(\aleph_1)\): A variation of Fraenkel's model, \(\cal N1\) | Historical notes
Description: Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false
Parameter(s): This model does not depend on parameters
All Forms Known to be True in \(\cal N12(\aleph_1)\):
423,
421,
420,
419,
418,
412,
411,
404,
403,
402,
401,
400,
398,
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,
337,
336-n,
330,
328,
327,
325,
324,
322,
315,
313,
309,
307,
306,
305,
304,
296,
294,
289,
288-n,
282,
280,
279,
273,
272,
260,
252,
251,
250,
249,
243,
231,
223,
222,
221,
217,
216,
212,
211,
209,
208,
207-alpha,
206,
203,
202,
199(\(n\)),
198,
197,
196-alpha,
194,
191,
190,
189,
187,
186,
185,
182,
181,
170,
169,
167,
165,
155,
154,
151,
150,
145,
142,
139,
137-k,
132,
131,
130,
128,
127,
126,
124,
122,
121,
120-K,
119,
113,
111,
108,
106,
104,
98,
94,
93,
92,
91,
86-alpha,
84,
83,
82,
80,
79,
78,
77,
74,
73,
70,
64,
63,
58,
57,
48-K,
47-n,
43,
42,
40,
39,
38,
37,
35,
34,
33-n,
32,
31,
29,
27,
26,
25,
24,
23,
19,
18,
17,
16,
13,
12,
11,
10,
9,
8,
6,
5,
0,
All Forms Known to be False in \(\cal N12(\aleph_1)\):
430-p,
427,
409,
408,
407,
399,
391,
384,
359,
347,
345,
344,
343,
335-n,
334,
333,
332,
331,
326,
323,
317,
303,
295,
284,
270,
239,
218,
214,
213,
201,
193,
192,
168,
163,
152,
149,
133,
123,
109,
107,
101,
100,
95-F,
88,
85,
68,
67,
66,
62,
61,
60,
50,
49,
45-n,
44,
41,
36,
30,
28-p,
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: 23-63-187-191-202
Falses that are implied by others list: 3-41-45-67-88-133-152-163-192-193-347
References for models trues falses list: See references for \(\cal N1\), A.~Rubin [1978], Higasikawa [1995], Jech [1973b] (pp 127and 131), Truss [1978], Brunner [1982a], Blass [1979], notes 18, 60, 108, and 120(4, 34, 46, and 56).
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