Fraenkel \(\cal N6\): Levy's Model I | Historical notes
Description: \(A=\{a_n : n\in\omega\}\) and \(A = \bigcup \{P_n: n\in\omega\}\), where \(P_0 = \{a_0\}\), \(P_1 = \{a_1,a_2\}\), \(P_2 =\{a_3,a_4,a_5\}\), \(P_3 = \{a_6,a_7,a_8,a_9,a_{10}\}\), \(\cdots\); in generalfor \(n>0\), \(|P_n| = p_n\), where \(p_n\) is the \(n\)th prime
Parameter(s): This model does not depend on parameters
All Forms Known to be True in \(\cal N6\):
430-p,
425,
423,
420,
419,
418,
416,
415,
414,
413,
397,
396,
395,
390,
389,
387,
382,
381,
375,
374-n,
373-n,
371,
369,
368,
367,
366,
364,
363,
362,
361,
350,
349,
342-n,
337,
336-n,
333,
330,
329,
328,
316,
315,
313,
309,
307,
306,
305,
294,
292,
289,
288-n,
287,
280,
276,
273,
272,
269,
268,
254,
252,
251,
250,
232,
223,
222,
221,
218,
212,
211,
208,
206,
203,
199(\(n\)),
197,
194,
191,
190,
189,
185,
182,
173,
170,
169,
155,
147,
145,
144,
142,
140,
139,
137-k,
131,
130,
127,
126,
125,
120-K,
119,
118,
116,
115,
114,
112,
111,
108,
106,
104,
95-F,
94,
93,
92,
91,
90,
89,
88,
84,
83,
82,
80,
79,
78,
77,
76,
74,
73,
70,
67,
64,
61,
58,
52,
51,
48-K,
47-n,
46-K,
45-n,
38,
37,
35,
34,
33-n,
25,
19,
18,
13,
12,
11,
6,
5,
0,
All Forms Known to be False in \(\cal N6\):
426,
421,
412,
411,
410,
409,
408,
407,
406,
403,
402,
401,
400,
399,
398,
394,
393,
392,
391,
386,
385,
384,
380,
379,
378,
377,
376,
359,
358,
355,
354,
352,
347,
345,
344,
343,
341,
340,
338,
334,
332,
331,
327,
325,
324,
323,
322,
317,
314,
308-p,
303,
302,
295,
286,
270,
264,
262,
261,
260,
259,
258,
257,
256,
255,
231,
214,
213,
202,
192,
181,
174-alpha,
171,
168,
165,
164,
161,
154,
153,
152,
151,
150,
133,
132,
129,
123,
122,
121,
113,
107,
101,
100,
87-alpha,
86-alpha,
85,
71-alpha,
68,
62,
60,
50,
49,
44,
43,
41,
40,
39,
36,
32,
31,
30,
29,
27,
23,
21,
20,
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: 191-218
Falses that are implied by others list: 164-171-308-314-334-358-379
References for models trues falses list: References Bleicher [1965],Brunner [1984b],Degen [1988], Howard/Yorke [1987], Jech [1973b] (Theorem 7.11 and prob 7.15), Keremedis [1996a],Levy [1962], H.~Rubin/J.~Rubin [1985], Shannon [1990], notes 18, 35, 95, and 120(2 and 56).
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