Fraenkel \(\cal N5\): The Mathias/Pincus Model II (an extension of \(\cal N4\)) | Historical notes
Description: \(A\) iscountably infinite; \(\precsim\) and \(\le\) are universal homogeneous partialand linear orderings, respectively, on \(A\), (See Jech [1973b]p101 for definitions.); \(\cal G\) is the group of all order automorphismson \((A,\precsim,\le)\); 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 \(\cal N5\):
423,
421,
420,
419,
401,
390,
389,
387,
380,
378,
377,
374-n,
373-n,
371,
369,
368,
367,
366,
364,
363,
362,
361,
358,
357,
356,
352,
350,
349,
342-n,
338,
336-n,
330,
329,
327,
324,
323,
315,
314,
313,
309,
308-p,
307,
306,
305,
294,
293,
289,
288-n,
285,
283,
280,
276,
273,
272,
271-n,
270,
269,
268,
252,
251,
250,
249,
231,
223,
222,
216,
213,
212,
211,
209,
207-alpha,
206,
203,
199(\(n\)),
198,
197,
194,
191,
190,
189,
182,
178-n-N,
170,
169,
165,
151,
150,
146,
145,
142,
140,
139,
137-k,
132,
130,
127,
122,
121,
120-K,
119,
111,
108,
104,
102,
94,
93,
92,
91,
88,
85,
83,
80,
79,
74,
73,
70,
64,
62,
61,
60,
48-K,
47-n,
46-K,
45-n,
38,
37,
35,
34,
33-n,
32,
31,
30,
27,
26,
25,
24,
23,
19,
18,
16,
13,
12,
11,
10,
6,
5,
0,
All Forms Known to be False in \(\cal N5\):
430-p,
427,
426,
407,
391,
388,
384,
359,
347,
346,
345,
335-n,
334,
333,
328,
317,
316,
303,
292,
286,
264,
262,
261,
260,
259,
258,
257,
256,
255,
239,
218,
214,
202,
193,
192,
188,
185,
181,
179-epsilon,
174-alpha,
168,
161,
152,
149,
144,
133,
131,
129,
126,
118,
115,
114,
113,
112,
109,
106,
101,
100,
99,
95-F,
90,
89,
87-alpha,
86-alpha,
84,
82,
77,
76,
71-alpha,
67,
66,
51,
50,
49,
44,
43,
41,
40,
39,
28-p,
20,
15,
14,
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: 16-23-24-30-60-91-191
Falses that are implied by others list: 49-84-99-118-131-144
References for models trues falses list: References Krom [1986], Mathias [1967],Pincus [1969], Felgner/Jech [1973], Jech [1973b], Howard [1973], Howard/Rubin [1995b],Levy [1958], notes 2(2,8,9), 18, 33, 94, and 120(28, 34 and 49).
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