Cohen \(\cal M18\): Shelah's Model I | Historical notes
Description: Shelah modified Solovay's model, \(\cal M5\), and constructed a model without using an inaccessible cardinal in which the Principle of Dependent Choices (Form 43) is true and every set of reals has the property of Baire (Form142 is false)
Parameter(s): This model does not depend on parameters
All Forms Known to be True in \(\cal M18\):
423,
420,
419,
418,
412,
411,
404,
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,
325,
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,
93,
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 \(\cal M18\):
430-p,
427,
409,
408,
407,
399,
391,
385,
384,
345,
344,
343,
335-n,
334,
333,
332,
331,
326,
323,
317,
313,
311,
310,
303,
295,
292,
287,
284,
270,
264,
239,
225,
222,
221,
218,
213,
202,
201,
164,
163,
149,
147,
142,
133,
130,
123,
114,
112,
109,
107,
99,
95-F,
91,
90,
89,
88,
85,
79,
70,
68,
67,
66,
63,
62,
61,
60,
52,
50,
49,
36,
30,
28-p,
15,
14,
1,
A minimial list of forms whose truth in this model imply all others that are true in this model: 43-93
Falses that are implied by others list: 142-163
References for models trues falses list: ReferencesShelah [1984], Solovay [1970], Stern [1985],and Brunner [1982a].
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