22:
272:
143:. Given that the relevant cardinals exist, it is consistent with ZFC either that the first measurable cardinal is strongly compact, or that the first strongly compact cardinal is supercompact; these cannot both be true, however. A measurable limit of strongly compact cardinals is strongly compact, but the least such limit is not supercompact.
150:. Some set theorists conjecture that existence of a strongly compact cardinal is equiconsistent with that of a supercompact cardinal. However, a proof is unlikely until a canonical inner model theory for supercompact cardinals is developed.
127:
The property of strong compactness may be weakened by only requiring this compactness property to hold when the original collection of statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness. A cardinal κ is
124:
property of finitary logic. Specifically, a statement which follows from some other collection of statements should also follow from some subcollection having cardinality less than κ.
120:κ is defined by requiring the number of operands for each operator to be less than κ; then κ is strongly compact if its logic satisfies an analog of the
51:
313:
105:
An uncountable cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-complete ultrafilter.
252:
172:
73:
44:
201:
306:
245:
Set Theory: An
Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76)
337:
34:
332:
287:
38:
30:
299:
129:
55:
140:
8:
160:
136:
121:
210:
248:
220:
117:
113:
109:
132:
if and only if it is κ-compact; this was the original definition of that concept.
193:
147:
157:
which holds for an inaccessible cardinal if and only if it is strongly compact.
283:
99:
224:
326:
154:
146:
The consistency strength of strong compactness is strictly above that of a
91:
279:
87:
215:
108:
Strongly compact cardinals were originally defined in terms of
271:
116:
are allowed to take infinitely many operands. The logic on a
194:"The super tree property at the successor of a singular"
324:
191:
163:is a second-order analog of strong compactness.
43:but its sources remain unclear because it lacks
307:
242:
192:Hachtman, Sherwood; Sinapova, Dima (2020).
314:
300:
214:
74:Learn how and when to remove this message
325:
266:
15:
13:
14:
349:
173:List of large cardinal properties
270:
20:
153:Jech obtained a variant of the
185:
1:
235:
202:Israel Journal of Mathematics
286:. You can help Knowledge by
178:
7:
166:
135:Strong compactness implies
10:
354:
265:
225:10.1007/s11856-020-2000-5
96:strongly compact cardinal
247:. Elsevier Science Ltd.
29:This article includes a
58:more precise citations.
282:-related article is a
243:Drake, F. R. (1974).
98:is a certain kind of
139:, and is implied by
31:list of references
295:
294:
114:logical operators
84:
83:
76:
345:
338:Set theory stubs
316:
309:
302:
274:
267:
258:
229:
228:
218:
198:
189:
141:supercompactness
118:regular cardinal
110:infinitary logic
79:
72:
68:
65:
59:
54:this article by
45:inline citations
24:
23:
16:
353:
352:
348:
347:
346:
344:
343:
342:
333:Large cardinals
323:
322:
321:
320:
263:
261:
255:
238:
233:
232:
196:
190:
186:
181:
169:
148:Woodin cardinal
80:
69:
63:
60:
49:
35:related reading
25:
21:
12:
11:
5:
351:
341:
340:
335:
319:
318:
311:
304:
296:
293:
292:
275:
260:
259:
253:
239:
237:
234:
231:
230:
209:(1): 473–500.
183:
182:
180:
177:
176:
175:
168:
165:
130:weakly compact
100:large cardinal
90:, a branch of
82:
81:
64:September 2023
39:external links
28:
26:
19:
9:
6:
4:
3:
2:
350:
339:
336:
334:
331:
330:
328:
317:
312:
310:
305:
303:
298:
297:
291:
289:
285:
281:
276:
273:
269:
268:
264:
256:
254:0-444-10535-2
250:
246:
241:
240:
226:
222:
217:
212:
208:
204:
203:
195:
188:
184:
174:
171:
170:
164:
162:
161:Extendibility
158:
156:
155:tree property
151:
149:
144:
142:
138:
137:measurability
133:
131:
125:
123:
119:
115:
111:
106:
103:
101:
97:
93:
89:
78:
75:
67:
57:
53:
47:
46:
40:
36:
32:
27:
18:
17:
288:expanding it
277:
262:
244:
206:
200:
187:
159:
152:
145:
134:
126:
107:
104:
95:
85:
70:
61:
50:Please help
42:
122:compactness
92:mathematics
56:introducing
327:Categories
280:set theory
236:References
216:1806.00820
88:set theory
179:Footnotes
167:See also
112:, where
52:improve
251:
278:This
211:arXiv
197:(PDF)
37:, or
284:stub
249:ISBN
94:, a
221:doi
207:236
86:In
329::
219:.
205:.
199:.
102:.
41:,
33:,
315:e
308:t
301:v
290:.
257:.
227:.
223::
213::
77:)
71:(
66:)
62:(
48:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.