Fraenkel \(\cal N2\): The Second Fraenkel Model | Historical notes
Description: The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).
Parameter(s): This model does not depend on parameters
All Forms Known to be True in \(\cal N2\):
425,
420,
419,
418,
416,
415,
414,
413,
397,
396,
395,
390,
389,
387,
382,
381,
375,
371,
369,
368,
367,
366,
364,
363,
362,
361,
350,
349,
337,
334,
330,
329,
328,
316,
315,
313,
309,
307,
306,
305,
294,
292,
289,
287,
280,
273,
272,
254,
252,
251,
232,
225,
223,
222,
221,
212,
211,
208,
206,
203,
199(\(n\)),
197,
194,
191,
190,
189,
185,
182,
173,
170,
169,
163,
155,
147,
145,
144,
142,
139,
137-k,
131,
130,
127,
126,
125,
119,
118,
116,
115,
114,
112,
108,
106,
104,
99,
94,
93,
92,
91,
90,
89,
84,
83,
82,
79,
78,
77,
76,
74,
70,
67,
65,
64,
58,
52,
51,
38,
37,
35,
34,
25,
19,
13,
6,
5,
0,
All Forms Known to be False in \(\cal N2\):
426,
422-n,
421,
412,
411,
410,
409,
408,
407,
406,
403,
402,
401,
400,
399,
398,
394,
393,
392,
391,
388,
386,
385,
384,
380,
379,
378,
377,
376,
359,
358,
355,
354,
352,
347,
346,
345,
344,
343,
341,
340,
338,
335-n,
333,
332,
331,
327,
326,
325,
324,
323,
322,
317,
303,
302,
295,
286,
284,
270,
264,
262,
261,
260,
259,
258,
257,
256,
255,
250,
239,
231,
218,
217,
216,
215,
214,
213,
202,
201,
198,
192,
181,
174-alpha,
168,
167,
166,
165,
164,
163,
161,
154,
153,
152,
151,
150,
133,
132,
129,
128,
123,
122,
121,
113,
111,
110,
109,
107,
101,
100,
98,
95-F,
88,
87-alpha,
86-alpha,
85,
80,
71-alpha,
68,
66,
62,
61,
60,
50,
49,
45-n,
44,
43,
41,
40,
39,
36,
32,
31,
30,
29,
27,
23,
21,
20,
18,
17,
16,
15,
14,
10,
9,
8,
7,
4,
3,
2,
1,
A minimial list of forms whose truth in this model imply all others that are true in this model: 65-99-163-191-334
Falses that are implied by others list: 18-45-98-128-163-164-198-216-388
References for models trues falses list: References Brunner [1982a],\cite{1983c}, \cite{1984b}, \cite{1984f}, De la Cruz, Hall, Howard,Keremedis, Rubin [2002a], Fraenkel [1922], Hickman [1978b], Howard [1984b], Jech [1973b],Keremedis [1999a], Pincus [1972a],\cite{1972b}, notes18, 93, 95, 105, and 120(2 and 56).
Back