Cohen \(\cal M5(\aleph)\): Solovay's Model | Historical notes
Description: An inaccessible cardinal \(\aleph\) is collapsed to \(\aleph_1\) in the outer model and then \(\cal M5(\aleph)\) is the smallest model containing the ordinals and \(\Bbb R\)
Parameter(s): This model depends on the following parameter(s): \(\alpha\), \(\alpha\): with \( \aleph = \aleph_{\alpha}\) an inaccessible cardinal
All Forms Known to be True in \(\cal M5(\aleph)\):
423,
420,
419,
418,
412,
411,
404,
390,
389,
388,
387,
380,
379,
378,
377,
376,
375,
374-n,
373-n,
369,
362,
361,
358,
357,
355,
354,
353,
352,
351,
350,
342-n,
341,
340,
339,
338,
336-n,
325,
307,
304,
296,
288-n,
282,
279,
249,
243,
217,
216,
211,
209,
199(\(n\)),
198,
194,
185,
182,
167,
155,
154,
150,
132,
131,
128,
127,
126,
124,
119,
113,
108,
106,
104,
98,
94,
84,
83,
82,
80,
78,
77,
74,
73,
64,
57,
43,
42,
38,
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 \(\cal M5(\aleph)\):
430-p,
427,
409,
408,
407,
399,
394,
392,
391,
385,
384,
368,
367,
366,
363,
359,
347,
345,
344,
343,
337,
335-n,
334,
333,
332,
331,
326,
323,
320,
319,
318,
317,
313,
312,
311,
310,
309,
306,
305,
303,
302,
295,
292,
287,
286,
284,
281,
280,
277,
272,
270,
264,
262,
261,
260,
259,
258,
257,
256,
255,
252,
251,
239,
234,
225,
224,
222,
221,
218,
213,
203,
202,
201,
195,
184,
170,
169,
168,
164,
163,
149,
147,
142,
133,
130,
123,
117,
114,
112,
109,
107,
101,
100,
99,
95-F,
93,
92,
91,
90,
89,
88,
85,
79,
70,
68,
67,
66,
63,
62,
61,
60,
52,
51,
50,
49,
40,
36,
30,
28-p,
20,
15,
14,
1,
A minimial list of forms whose truth in this model imply all others that are true in this model: 43-307-369
Falses that are implied by others list: 93-163-169-224-234-277-280-281-312-318
References for models trues falses list: References Solovay [1965], \cite{1970}, Brunner [1982a], Jech [1968a], Kanovei [1991], Monro [1973a],Mathias [1968], Prikry [1976], Raisonnier [1984], Shelah [1984], Stern [1979], Truss [1974b], Wright [1973], Wagon [1985], and note20.
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