164:
for each measurable cardinal, C minus ∪S is finite. Note that every κ \ C is either finite or Prikry generic for K at κ except for members of C below a measurable cardinal below κ. For every uncountable set x of ordinals, there is y ∈ K such that x ⊂ y and |x| = |y|.
163:
If K has no inaccessible limit of measurable cardinals and no proper class of measurable cardinals, then there is a maximal and unique (except for a finite set of ordinals) set C (called a system of indiscernibles) for K such that for every sequence S in K of measure one sets consisting of one set
192:. For example, if K does not have overlapping total extenders, and κ is singular strong limit, and 2 = κ, then κ has Mitchell order at least κ in K. Conversely, a failure of the singular cardinal hypothesis can be obtained (in a generic extension) from κ with o(κ) = κ.
187:
For core models without overlapping total extenders, the systems of indiscernibles are well understood. Although (if K has an inaccessible limit of measurable cardinals), the system may depend on the set to be covered, it is well-determined and unique in a weaker sense. One application of the
159:
If K has only one measurable cardinal κ, then for every uncountable set x of ordinals, there is y ∈ K such that x ⊂ y and |x| = |y|. Here C is either empty or Prikry generic over K (so it has order type ω and is cofinal in κ) and unique except up to a finite
145:-Erdős cardinals, then for a particular countable (in K) and definable in K sequence of functions from ordinals to ordinals, every set of ordinals closed under these functions is a union of a countable number of sets in K. If L=K, these are simply the primitive recursive functions.
195:
For core models with overlapping total extenders (that is with a cardinal strong up to a measurable one), the systems of indiscernibles are poorly understood, and applications (such as the weak covering) tend to avoid rather than analyze the indiscernibles.
49:. A covering lemma asserts that under some particular anti-large cardinal assumption, the core model exists and is maximal in a sense that depends on the chosen large cardinal. The first such result was proved by
167:
For every uncountable set x of ordinals, there is a set C of indiscernibles for total extenders on K such that there is y ∈ K and x ⊂ y and |x| = |y|.
234:
204:
If K exists, then every regular Jónsson cardinal is Ramsey in K. Every singular cardinal that is regular in K is measurable in K.
207:
Also, if the core model K(X) exists above a set X of ordinals, then it has the above discussed covering properties above X.
188:
covering is counting the number of (sequences of) indiscernibles, which gives optimal lower bounds for various failures of the
189:
62:
17:
252:
23:
262:
54:
43:
74:
230:
222:
257:
31:
246:
50:
226:
35:
170:
K computes the successors of singular and weakly compact cardinals correctly (
118:
58:
39:
42:, that is, in a sense, maximal and approximates the structure of the
156: ∈ K such that x ⊂ y and |x| = |y|.
148:
If K has no measurable cardinals, then for every uncountable set
137:
If the core model K exists (and has no Woodin cardinals), then
30:is used to prove that the non-existence of certain
217:Mitchell, William (2010), "The covering lemma",
73:For example, if there is no inner model for a
8:
178:, then cofinality((κ)) ≥ |κ|.
61:does not exist, which is now known as
34:leads to the existence of a canonical
174:). Moreover, if |κ| > ω
7:
85:, that is for every uncountable set
81:is the core model and satisfies the
77:, then the Dodd–Jensen core model,
14:
221:, Springer, pp. 1497–1594,
1:
190:singular cardinals hypothesis
227:10.1007/978-1-4020-5764-9_19
183:Extenders and indiscernibles
105:has the same cardinality as
279:
24:foundations of mathematics
15:
63:Jensen's covering theorem
18:Jensen's covering theorem
219:Handbook of Set Theory
172:Weak Covering Property
152:of ordinals, there is
89:of ordinals, there is
55:constructible universe
200:Additional properties
121:does not exist, then
44:von Neumann universe
75:measurable cardinal
253:Inner model theory
236:978-1-4020-4843-2
83:covering property
270:
239:
160:initial segment.
278:
277:
273:
272:
271:
269:
268:
267:
263:Covering lemmas
243:
242:
237:
216:
213:
202:
185:
177:
144:
135:
71:
32:large cardinals
20:
12:
11:
5:
276:
274:
266:
265:
260:
255:
245:
244:
241:
240:
235:
212:
209:
201:
198:
184:
181:
180:
179:
175:
168:
165:
161:
157:
146:
142:
134:
131:
70:
67:
28:covering lemma
13:
10:
9:
6:
4:
3:
2:
275:
264:
261:
259:
256:
254:
251:
250:
248:
238:
232:
228:
224:
220:
215:
214:
210:
208:
205:
199:
197:
193:
191:
182:
173:
169:
166:
162:
158:
155:
151:
147:
141:If K has no ω
140:
139:
138:
132:
130:
128:
125: =
124:
120:
116:
113: ∈
112:
108:
104:
100:
97: ⊃
96:
92:
88:
84:
80:
76:
68:
66:
64:
60:
56:
52:
51:Ronald Jensen
48:
45:
41:
38:, called the
37:
33:
29:
25:
19:
218:
206:
203:
194:
186:
171:
153:
149:
136:
126:
122:
114:
110:
106:
102:
98:
94:
90:
86:
82:
78:
72:
46:
27:
21:
36:inner model
247:Categories
211:References
93:such that
40:core model
16:See also:
57:assuming
133:Versions
53:for the
69:Example
22:In the
258:Lemmas
233:
117:. (If
109:, and
231:ISBN
26:, a
223:doi
129:.)
249::
229:,
101:,
65:.
225::
176:1
154:y
150:x
143:1
127:L
123:K
119:0
115:K
111:y
107:x
103:y
99:x
95:y
91:y
87:x
79:K
59:0
47:V
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.