Cohen \(\cal M1(\langle\omega_1\rangle)\): Cohen/Pincus Model | Historical notes
Description: Pincus extends the methods of Cohen and adds a generic \(\omega_1\)-sequence, \(\langle I_{\alpha}: \alpha\in\omega_1\rangle\), of denumerable sets, where \(I_0\) is a denumerable set of generic reals, each \(I_{\alpha+1}\) is a generic set of enumerations of \(I_{\alpha}\), and for a limit ordinal \(\lambda\),\(I_{\lambda}\) is a generic set of choice functions for \(\{I_{\alpha}:\alpha \le \lambda\}\)
Parameter(s): This model does not depend on parameters
All Forms Known to be True in \(\cal M1(\langle\omega_1\rangle)\):
425,
423,
420,
419,
418,
412,
411,
405,
404,
401,
390,
389,
388,
387,
380,
379,
378,
377,
376,
375,
374-n,
373-n,
362,
361,
358,
357,
355,
354,
353,
352,
351,
350,
342-n,
341,
340,
339,
338,
336-n,
327,
325,
308-p,
304,
296,
295,
293,
288-n,
285,
283,
282,
280,
279,
276,
269,
268,
250,
249,
243,
217,
216,
214,
211,
209,
199(\(n\)),
198,
194,
185,
182,
178-n-N,
173,
167,
155,
154,
152,
150,
146,
142,
140,
132,
131,
128,
127,
126,
124,
122,
121,
120-K,
119,
113,
111,
108,
106,
104,
102,
98,
97,
94,
93,
88,
84,
83,
82,
80,
78,
77,
76,
75,
74,
73,
64,
62,
61,
57,
48-K,
47-n,
46-K,
45-n,
43,
42,
38,
35,
34,
33-n,
32,
31,
30,
29,
27,
26,
24,
19,
18,
17,
16,
13,
12,
11,
10,
9,
8,
6,
5,
4,
0,
All Forms Known to be False in \(\cal M1(\langle\omega_1\rangle)\):
430-p,
427,
391,
359,
347,
335-n,
334,
333,
329,
328,
292,
286,
264,
262,
261,
260,
259,
258,
257,
256,
255,
239,
231,
218,
213,
202,
168,
164,
163,
151,
149,
147,
133,
114,
112,
109,
101,
100,
95-F,
91,
90,
89,
85,
67,
66,
60,
44,
40,
39,
28-p,
23,
21,
20,
1,
A minimial list of forms whose truth in this model imply all others that are true in this model: 43-97-214-295
Falses that are implied by others list: 39-85-91-151-163-328-329
References for models trues falses list: ReferencesPincus [1977a], \cite{1997}, Brunner [1982a], Note 18,and Note 120(32, 43, and 47).
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