2750:
2189:
1975:
3184:
1877:
2088:
1172:
2473:
2589:
2557:
2876:
3124:
2957:
3210:
3003:
2645:
2499:
2051:
3527:
3305:
2619:
660:
2000:
1915:
3483:
3408:
3086:
3055:
2525:
594:
546:
3252:
2924:
2405:
2366:
2275:
2224:
1829:
1258:
1215:
622:
2335:
1221:
is ordered by inclusion (i.e., subtrees are stronger conditions). The intersection of all trees in the generic filter defines a countable sequence which is cofinal in ω
2683:
3350:
3330:
3277:
3029:
2977:
2902:
2020:
379:
by any cardinal κ so construct a model where the continuum has size at least κ. Here, there is no restriction. If κ has cofinality ω, the reals end up bigger than κ.
3434:
2435:
3461:
3377:
1630:
Radin forcing (after Lon Berk Radin), a technically involved generalization of
Magidor forcing, adds a closed, unbounded subset to some regular cardinal λ.
3230:
2849:
2678:
2312:
2117:
2122:
939:
is the set of all functions from a subset of κ of cardinality less than κ to λ (for fixed cardinals κ and λ). Forcing with this poset collapses λ down to κ.
58:
used will determine what statements hold in the new universe (the 'extension'); to force a statement of interest thus requires construction of a suitable
3593:
3129:
1920:
992:, one of the best known is a generalization of Prikry forcing used to change the cofinality of a cardinal to a given smaller regular cardinal.
930:
is the set of all finite sequences of ordinals less than a given cardinal λ. If λ is uncountable then forcing with this poset collapses λ to ω.
3554:
Schlindwein, C., Shelah's work on non-semiproper iterations, I, Archive for
Mathematical Logic, vol. 47, no. 6, pp. 579 -- 606 (2008)
372:
distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum hypothesis.
1834:
1270:
proved that if CH holds then a generic object of Namba forcing does not exist in the generic extension by Namba', and vice versa.
3623:
3602:
3579:
1136:
399:
Hechler forcing (after
Stephen Herman Hechler) is used to show that Martin's axiom implies that every family of less than
2440:
17:
1732:
is a set of finite sequences containing all initial segments of its members, and is called perfect if for any element
2562:
2056:
2530:
920:
These posets will collapse various cardinals, in other words force them to be equal in size to smaller cardinals.
2855:
3664:
3091:
712:
to handle semi-proper forcings, such as Prikry forcing, and generalizations, notably including Namba forcing.
2929:
728:
are countable, is consistent with ZFC. (Borel's conjecture is not consistent with the continuum hypothesis.)
629:
1119:
3189:
2982:
2624:
2478:
3488:
3282:
2598:
334:
635:
119:
2025:
1890:
3466:
3034:
2504:
567:
519:
3235:
2907:
2745:{\displaystyle \{\alpha \,\colon (\exists \beta )(\langle \alpha ,\beta \rangle \in \bigcup G)\}}
2383:
2344:
2253:
2202:
1807:
1236:
1193:
725:
600:
425:
is a finite sequence of natural numbers (considered as functions from a finite ordinal to ω) and
2317:
1360:. This forcing notion can be used to change to cofinality of κ while preserving all cardinals.
701:
689:
685:
1980:
1368:
Taking a product of forcing conditions is a way of simultaneously forcing all the conditions.
3382:
3335:
3060:
3008:
2962:
2852:
2005:
1638:
28:
3413:
2414:
3647:
3439:
3355:
2231:
1233:
such that there is a node below which the ordering is linear and above which each node has
972:. This poset collapses all cardinals less than λ onto κ, but keeps λ as the successor to κ.
3633:
8:
1634:
693:
626:
3310:
3257:
2882:
1174:(nonempty downward closed subsets of the set of finite sequences of ordinals less than ω
708:, who introduced proper forcing. Revised countable support iteration was introduced by
3215:
2834:
2654:
2288:
2184:{\displaystyle \bigcup \{\sigma \,\colon (\exists C)(\langle \sigma ,C\rangle \in G)\}}
2093:
915:
681:
3619:
3598:
3575:
2235:
557:
3629:
1263:
989:
696:
at regular cardinals. Iterated forcing with countable support was investigated by
671:
368:
This poset satisfies the countable chain condition. Forcing with this poset adds ω
3571:
2828:
2816:
1789:
1650:
1267:
1103:
903:
709:
705:
677:
1614:
and such that for every regular cardinal γ the number of elements α of γ with
3658:
3611:
3588:
2411:
equal to the set of finite sets of pairs of countable ordinals, such that if
2227:
721:
697:
553:
328:
3179:{\displaystyle x\in p\Leftrightarrow V_{\beta }\models \varphi (\gamma ,x)}
1777:
597:
549:
316:
2768:) is the set of all those partial functions from the natural numbers into
3563:
2765:
1970:{\displaystyle \langle \sigma ',C'\rangle \leq \langle \sigma ,C\rangle }
1633:
If λ is a sufficiently large cardinal, then the forcing keeps λ regular,
977:
388:
1114:
Namba forcing (after Kanji Namba) is used to change the cofinality of ω
36:
3545:
Shelah, S., Proper and
Improper Forcing (Claim XI.4.2), Springer, 1998
403:
functions from ω to ω is eventually dominated by some such function.
3307:(the conditions being the minimal representations of elements of
2788:
is a subset of the natural numbers with infinite complement, and
2772:
whose domain is coinfinite; or equivalently the set of all pairs
746:
is a subset of the finite sequences of natural numbers such that
3618:, Studies in Logic, vol. 34, London: College Publications,
2368:
is preserved, and if CH holds then all cardinals are preserved.
1724:
is the set of all perfect trees contained in the set of finite
1872:{\displaystyle P=\{\langle \sigma ,C\rangle \,\colon \sigma }
692:
introduced another type of iterated forcing to determine the
44:
2241:
387:
Grigorieff forcing (after Serge
Grigorieff) destroys a free
1659:
is the set of Borel subsets of of positive measure, where
2819:, and is minimal with respect to reals (but not minimal).
2371:
1298:
is a finite subset of a fixed measurable cardinal κ, and
3570:, Springer Monographs in Mathematics, Berlin, New York:
1482:, each with a largest element 1 is the set of functions
676:
Iterated forcing with finite supports was introduced by
700:
in his proof of the consistency of Borel's conjecture,
3491:
3469:
3442:
3416:
3385:
3358:
3338:
3313:
3285:
3260:
3238:
3218:
3192:
3132:
3094:
3063:
3037:
3011:
2985:
2965:
2932:
2910:
2885:
2858:
2837:
2686:
2657:
2627:
2601:
2565:
2533:
2507:
2481:
2443:
2417:
2386:
2347:
2320:
2291:
2256:
2205:
2125:
2096:
2059:
2028:
2008:
1983:
1923:
1893:
1837:
1810:
1239:
1196:
1139:
724:
to show that Borel's conjecture, which says that all
638:
603:
570:
522:
3332:). This poset is the Vopenka forcing for subsets of
2281:
equal to the set of closed countable sequences from
1167:{\displaystyle T\subseteq \omega _{2}^{<\omega }}
135:
have the same cardinals (and the same cofinalities).
2468:{\displaystyle \langle \alpha ,\beta \rangle \in p}
298:if it is a filter that meets every dense subset of
3594:Set Theory: An Introduction to Independence Proofs
3521:
3477:
3455:
3428:
3402:
3371:
3344:
3324:
3299:
3271:
3246:
3224:
3204:
3178:
3118:
3080:
3049:
3023:
2997:
2971:
2951:
2918:
2896:
2870:
2843:
2744:
2672:
2639:
2613:
2583:
2551:
2519:
2493:
2467:
2429:
2399:
2360:
2329:
2306:
2269:
2218:
2183:
2111:
2082:
2045:
2014:
1994:
1969:
1909:
1871:
1823:
1564:(after William Bigelow Easton) of a set of posets
1252:
1209:
1166:
654:
616:
588:
540:
341:is the set of functions from a finite subset of ω
3656:
757:contains any initial sequence of any element of
735:is the set of Laver trees, ordered by inclusion.
3379:as the set of all representations for elements
2584:{\displaystyle \langle \gamma ,\delta \rangle }
2083:{\displaystyle \sigma '\subseteq \sigma \cup C}
844:has an infinite number of immediate successors
2552:{\displaystyle \langle \alpha ,\beta \rangle }
1675:then encodes a "random real": the unique real
1587:is a set of cardinals is the set of functions
1013:of natural numbers such that every element of
1688:. This real is "random" in the sense that if
625:(meaning the sets of paths through infinite,
90:is a countable transitive model of set theory
2739:
2724:
2712:
2687:
2578:
2566:
2546:
2534:
2456:
2444:
2226:is preserved. This method was introduced by
2178:
2166:
2154:
2129:
1964:
1952:
1946:
1924:
1904:
1859:
1847:
1844:
1684:in all rational intervals such that is in
945:If κ is regular and λ is inaccessible, then
319:, and adds a measure 1 set of random reals.
2871:{\displaystyle {\color {blue}{\text{HOD}}}}
988:Amongst many forcing notions developed by
66:that have been used in this construction.
3119:{\displaystyle (\beta ,\gamma ,\varphi )}
2796:into a fixed 2-element set. A condition
2693:
2242:Shooting a club with countable conditions
2135:
1862:
1776:. Forcing with perfect trees was used by
1692:is any subset of of measure 1, lying in
694:possible values of the continuum function
511:
2230:in order to show the consistency of the
1784:with minimal degree of constructibility.
1302:is an element of a fixed normal measure
127:is at most countable. This implies that
62:. This article lists some of the posets
2952:{\displaystyle {\mathcal {P}}(\alpha )}
1795:
1278:In Prikry forcing (after Karel Prikrý)
1009:of natural numbers and an infinite set
433:of functions from ω to ω. The element (
31:is a method of constructing new models
14:
3657:
2372:Shooting a club with finite conditions
704:, who introduced Axiom A forcing, and
3610:
3587:
2815:Silver forcing satisfies Fusion, the
2651:is ordered by reverse inclusion. In
2195:almost contained in each club set in
1005:is a pair consisting of a finite set
429:is a finite subset of some fixed set
382:
3562:
957:with domain of size less than κ and
2831:) is used to generically add a set
1178:) which have the property that any
665:
556:to prove, among other results, the
315:Amoeba forcing is forcing with the
24:
3205:{\displaystyle x\subseteq \alpha }
2998:{\displaystyle A\subseteq \alpha }
2935:
2860:
2822:
2700:
2640:{\displaystyle \delta <\alpha }
2494:{\displaystyle \alpha \leq \beta }
2349:
2207:
2142:
1363:
1241:
1198:
995:
983:
909:
572:
524:
394:
25:
3676:
3640:
3522:{\displaystyle A\in {\text{HOD}}}
3300:{\displaystyle P\in {\text{HOD}}}
2759:
2756:and all cardinals are preserved.
2614:{\displaystyle \beta <\gamma }
1644:
1463:: The product of a set of posets
1393:has the partial order defined by
1273:
934:Collapsing a cardinal to another:
375:More generally, one can replace ω
310:
3232:and its least representation is
2752:is a closed unbounded subset of
2337:is a closed unbounded subset of
2191:is a closed unbounded subset of
1887:is a closed unbounded subset of
1715:
1625:
1229:Namba' forcing is the subset of
1109:
765:is closed under initial segments
715:
322:
1520:for all but a finite number of
655:{\displaystyle 2^{<\omega }}
3568:Set Theory: Millennium Edition
3548:
3539:
3516:
3503:
3173:
3161:
3142:
3113:
3095:
3088:can be represented by a tuple
2946:
2940:
2736:
2709:
2706:
2697:
2667:
2661:
2301:
2295:
2175:
2151:
2148:
2139:
2106:
2100:
1383:are posets, the product poset
1017:is less than every element of
333:In Cohen forcing (named after
108:
13:
1:
3532:
2800:is stronger than a condition
2046:{\displaystyle C'\subseteq C}
1910:{\displaystyle \omega _{1}\}}
1102:Mathias forcing is named for
976:Levy collapsing is named for
3478:{\displaystyle {\text{HOD}}}
3050:{\displaystyle p\subseteq q}
2904:as the set of all non-empty
2520:{\displaystyle \alpha \in S}
902:Laver forcing satisfies the
589:{\displaystyle \Pi _{1}^{0}}
541:{\displaystyle \Pi _{1}^{0}}
7:
3247:{\displaystyle {\text{OD}}}
2919:{\displaystyle {\text{OD}}}
2400:{\displaystyle \omega _{1}}
2361:{\displaystyle \aleph _{1}}
2270:{\displaystyle \omega _{1}}
2219:{\displaystyle \aleph _{1}}
1824:{\displaystyle \omega _{1}}
1253:{\displaystyle \aleph _{2}}
1210:{\displaystyle \aleph _{2}}
925:Collapsing a cardinal to ω:
771:has a stem: a maximal node
684:to show the consistency of
617:{\displaystyle 2^{\omega }}
84:is the universe of all sets
69:
39:by adding a generic subset
10:
3681:
3212:. The translation between
1879:is a closed sequence from
1648:
913:
720:Laver forcing was used by
669:
326:
78:is a poset with order <
3279:is isomorphic to a poset
2926:subsets of the power set
2591:are distinct elements of
2330:{\displaystyle \bigcup G}
1334:is an initial segment of
1072:is an initial segment of
1021:. The order is defined by
761:, equivalently stated as
662:), ordered by inclusion.
120:countable chain condition
3005:, ordered by inclusion:
1995:{\displaystyle \sigma '}
1663:is called stronger than
1524:. The order is given by
1133:is the set of all trees
949:is the set of functions
726:strong measure zero sets
548:classes was invented by
3403:{\displaystyle p\in P'}
3345:{\displaystyle \alpha }
3081:{\displaystyle p\in P'}
3024:{\displaystyle p\leq q}
2972:{\displaystyle \alpha }
2827:Vopěnka forcing (after
2380:a stationary subset of
2250:a stationary subset of
2015:{\displaystyle \sigma }
1804:a stationary subset of
564:is the set of nonempty
96:is a generic subset of
3523:
3479:
3457:
3430:
3429:{\displaystyle A\in p}
3404:
3373:
3346:
3326:
3301:
3273:
3248:
3226:
3206:
3180:
3120:
3082:
3051:
3025:
2999:
2973:
2953:
2920:
2898:
2872:
2845:
2764:Silver forcing (after
2746:
2674:
2641:
2615:
2585:
2553:
2521:
2495:
2469:
2431:
2430:{\displaystyle p\in P}
2401:
2362:
2331:
2308:
2271:
2220:
2185:
2113:
2084:
2047:
2016:
1996:
1971:
1911:
1873:
1825:
1788:Sacks forcing has the
1667:if it is contained in
1260:immediate successors.
1254:
1217:immediate successors.
1211:
1168:
895:, uniquely determines
656:
618:
590:
542:
512:Jockusch–Soare forcing
345:× ω to {0,1} and
123:if every antichain in
3665:Forcing (mathematics)
3524:
3480:
3458:
3456:{\displaystyle G_{A}}
3431:
3405:
3374:
3372:{\displaystyle G_{A}}
3347:
3327:
3302:
3274:
3249:
3227:
3207:
3181:
3121:
3083:
3052:
3026:
3000:
2974:
2954:
2921:
2899:
2873:
2846:
2747:
2675:
2642:
2616:
2586:
2554:
2522:
2496:
2470:
2432:
2402:
2363:
2332:
2309:
2272:
2221:
2186:
2114:
2085:
2048:
2017:
1997:
1972:
1912:
1874:
1826:
1255:
1212:
1169:
657:
619:
591:
543:
191:is a nonempty subset
3489:
3467:
3440:
3414:
3383:
3356:
3336:
3311:
3283:
3258:
3236:
3216:
3190:
3130:
3092:
3061:
3035:
3009:
2983:
2963:
2930:
2908:
2883:
2856:
2835:
2684:
2655:
2625:
2599:
2563:
2531:
2505:
2479:
2441:
2415:
2384:
2345:
2318:
2289:
2254:
2232:continuum hypothesis
2203:
2123:
2094:
2057:
2026:
2006:
1981:
1921:
1891:
1835:
1808:
1796:Shooting a fast club
1282:is the set of pairs
1237:
1194:
1186:has an extension in
1137:
636:
601:
568:
520:
473:is in the domain of
409:is the set of pairs
2792:is a function from
1740:there is a segment
1728:sequences. (A tree
1163:
686:Suslin's hypothesis
585:
537:
441:) is stronger than
249:then there is some
3519:
3475:
3453:
3426:
3400:
3369:
3342:
3325:{\displaystyle P'}
3322:
3297:
3272:{\displaystyle P'}
3269:
3244:
3222:
3202:
3176:
3116:
3078:
3047:
3021:
2995:
2969:
2949:
2916:
2897:{\displaystyle P'}
2894:
2868:
2866:
2841:
2766:Jack Howard Silver
2742:
2670:
2637:
2611:
2581:
2549:
2517:
2491:
2465:
2427:
2397:
2358:
2327:
2304:
2267:
2216:
2181:
2109:
2080:
2043:
2012:
1992:
1967:
1907:
1869:
1821:
1780:to produce a real
1778:Gerald Enoch Sacks
1671:. The generic set
1306:on κ. A condition
1250:
1207:
1164:
1146:
1037:is stronger than
916:Collapsing algebra
652:
614:
586:
571:
538:
523:
383:Grigorieff forcing
18:Grigorieff forcing
3646:A.Miller (2009),
3625:978-1-84890-050-9
3604:978-0-444-86839-8
3581:978-3-540-44085-7
3501:
3473:
3295:
3242:
3225:{\displaystyle p}
3057:. Each condition
2914:
2864:
2844:{\displaystyle A}
2673:{\displaystyle V}
2307:{\displaystyle V}
2236:Suslin hypothesis
2112:{\displaystyle V}
1764:is stronger than
1461:Infinite products
1318:is stronger than
968:in the domain of
558:low basis theorem
16:(Redirected from
3672:
3649:Forcing Tidbits.
3636:
3607:
3584:
3555:
3552:
3546:
3543:
3528:
3526:
3525:
3520:
3515:
3514:
3502:
3499:
3484:
3482:
3481:
3476:
3474:
3471:
3462:
3460:
3459:
3454:
3452:
3451:
3435:
3433:
3432:
3427:
3409:
3407:
3406:
3401:
3399:
3378:
3376:
3375:
3370:
3368:
3367:
3351:
3349:
3348:
3343:
3331:
3329:
3328:
3323:
3321:
3306:
3304:
3303:
3298:
3296:
3293:
3278:
3276:
3275:
3270:
3268:
3253:
3251:
3250:
3245:
3243:
3240:
3231:
3229:
3228:
3223:
3211:
3209:
3208:
3203:
3185:
3183:
3182:
3177:
3154:
3153:
3125:
3123:
3122:
3117:
3087:
3085:
3084:
3079:
3077:
3056:
3054:
3053:
3048:
3030:
3028:
3027:
3022:
3004:
3002:
3001:
2996:
2978:
2976:
2975:
2970:
2958:
2956:
2955:
2950:
2939:
2938:
2925:
2923:
2922:
2917:
2915:
2912:
2903:
2901:
2900:
2895:
2893:
2877:
2875:
2874:
2869:
2867:
2865:
2862:
2850:
2848:
2847:
2842:
2783:
2771:
2751:
2749:
2748:
2743:
2679:
2677:
2676:
2671:
2646:
2644:
2643:
2638:
2620:
2618:
2617:
2612:
2590:
2588:
2587:
2582:
2558:
2556:
2555:
2550:
2526:
2524:
2523:
2518:
2500:
2498:
2497:
2492:
2474:
2472:
2471:
2466:
2436:
2434:
2433:
2428:
2406:
2404:
2403:
2398:
2396:
2395:
2367:
2365:
2364:
2359:
2357:
2356:
2336:
2334:
2333:
2328:
2313:
2311:
2310:
2305:
2276:
2274:
2273:
2268:
2266:
2265:
2225:
2223:
2222:
2217:
2215:
2214:
2190:
2188:
2187:
2182:
2118:
2116:
2115:
2110:
2089:
2087:
2086:
2081:
2067:
2052:
2050:
2049:
2044:
2036:
2021:
2019:
2018:
2013:
2001:
1999:
1998:
1993:
1991:
1976:
1974:
1973:
1968:
1945:
1934:
1916:
1914:
1913:
1908:
1903:
1902:
1878:
1876:
1875:
1870:
1830:
1828:
1827:
1822:
1820:
1819:
1772:is contained in
1727:
1710:
1620:
1613:
1582:
1552:
1533:
1519:
1508:
1481:
1456:
1440:
1424:
1392:
1359:
1350:is contained in
1342:is contained in
1329:
1317:
1293:
1259:
1257:
1256:
1251:
1249:
1248:
1216:
1214:
1213:
1208:
1206:
1205:
1173:
1171:
1170:
1165:
1162:
1154:
1097:
1088:is contained in
1067:
1048:
1036:
967:
963:
956:
943:Levy collapsing:
890:
875:, then the real
874:
858:
839:
829:
808:
798:
788:
672:Iterated forcing
666:Iterated forcing
661:
659:
658:
653:
651:
650:
623:
621:
620:
615:
613:
612:
595:
593:
592:
587:
584:
579:
547:
545:
544:
539:
536:
531:
499:
465:is contained in
457:is contained in
452:
420:
364:
354:
278:
268:
258:
248:
238:
228:
218:
208:
179:
169:
159:
27:In mathematics,
21:
3680:
3679:
3675:
3674:
3673:
3671:
3670:
3669:
3655:
3654:
3643:
3626:
3605:
3582:
3572:Springer-Verlag
3559:
3558:
3553:
3549:
3544:
3540:
3535:
3510:
3506:
3498:
3490:
3487:
3486:
3470:
3468:
3465:
3464:
3447:
3443:
3441:
3438:
3437:
3415:
3412:
3411:
3392:
3384:
3381:
3380:
3363:
3359:
3357:
3354:
3353:
3337:
3334:
3333:
3314:
3312:
3309:
3308:
3292:
3284:
3281:
3280:
3261:
3259:
3256:
3255:
3239:
3237:
3234:
3233:
3217:
3214:
3213:
3191:
3188:
3187:
3149:
3145:
3131:
3128:
3127:
3093:
3090:
3089:
3070:
3062:
3059:
3058:
3036:
3033:
3032:
3010:
3007:
3006:
2984:
2981:
2980:
2964:
2961:
2960:
2934:
2933:
2931:
2928:
2927:
2911:
2909:
2906:
2905:
2886:
2884:
2881:
2880:
2879:. Define first
2861:
2859:
2857:
2854:
2853:
2851:of ordinals to
2836:
2833:
2832:
2825:
2823:Vopěnka forcing
2773:
2769:
2762:
2685:
2682:
2681:
2680:, we have that
2656:
2653:
2652:
2626:
2623:
2622:
2600:
2597:
2596:
2564:
2561:
2560:
2532:
2529:
2528:
2527:, and whenever
2506:
2503:
2502:
2480:
2477:
2476:
2442:
2439:
2438:
2416:
2413:
2412:
2391:
2387:
2385:
2382:
2381:
2374:
2352:
2348:
2346:
2343:
2342:
2319:
2316:
2315:
2314:, we have that
2290:
2287:
2286:
2261:
2257:
2255:
2252:
2251:
2244:
2210:
2206:
2204:
2201:
2200:
2124:
2121:
2120:
2119:, we have that
2095:
2092:
2091:
2060:
2058:
2055:
2054:
2029:
2027:
2024:
2023:
2007:
2004:
2003:
1984:
1982:
1979:
1978:
1938:
1927:
1922:
1919:
1918:
1898:
1894:
1892:
1889:
1888:
1836:
1833:
1832:
1815:
1811:
1809:
1806:
1805:
1798:
1725:
1718:
1705:
1697:
1683:
1653:
1647:
1628:
1621:is less than γ.
1615:
1596:
1573:
1565:
1535:
1525:
1510:
1491:
1472:
1464:
1455:
1448:
1442:
1439:
1432:
1426:
1422:
1415:
1408:
1401:
1394:
1384:
1373:Finite products
1366:
1364:Product forcing
1351:
1319:
1307:
1283:
1276:
1244:
1240:
1238:
1235:
1234:
1224:
1201:
1197:
1195:
1192:
1191:
1177:
1155:
1150:
1138:
1135:
1134:
1125:
1117:
1112:
1089:
1080:is a subset of
1049:
1038:
1026:
998:
996:Mathias forcing
986:
984:Magidor forcing
965:
958:
954:
918:
912:
910:Levy collapsing
876:
868:
867:is generic for
853:
831:
821:
800:
790:
772:
718:
674:
668:
643:
639:
637:
634:
633:
608:
604:
602:
599:
598:
580:
575:
569:
566:
565:
532:
527:
521:
518:
517:
514:
482:
442:
410:
397:
395:Hechler forcing
385:
378:
371:
356:
346:
344:
331:
325:
313:
270:
260:
250:
240:
230:
220:
210:
200:
171:
161:
151:
111:
72:
23:
22:
15:
12:
11:
5:
3678:
3668:
3667:
3653:
3652:
3642:
3641:External links
3639:
3638:
3637:
3624:
3612:Kunen, Kenneth
3608:
3603:
3589:Kunen, Kenneth
3585:
3580:
3557:
3556:
3547:
3537:
3536:
3534:
3531:
3518:
3513:
3509:
3505:
3497:
3494:
3450:
3446:
3425:
3422:
3419:
3398:
3395:
3391:
3388:
3366:
3362:
3341:
3320:
3317:
3291:
3288:
3267:
3264:
3221:
3201:
3198:
3195:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3152:
3148:
3144:
3141:
3138:
3135:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3076:
3073:
3069:
3066:
3046:
3043:
3040:
3020:
3017:
3014:
2994:
2991:
2988:
2968:
2948:
2945:
2942:
2937:
2892:
2889:
2840:
2824:
2821:
2817:Sacks property
2761:
2760:Silver forcing
2758:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2692:
2689:
2669:
2666:
2663:
2660:
2636:
2633:
2630:
2610:
2607:
2604:
2580:
2577:
2574:
2571:
2568:
2548:
2545:
2542:
2539:
2536:
2516:
2513:
2510:
2490:
2487:
2484:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2426:
2423:
2420:
2394:
2390:
2373:
2370:
2355:
2351:
2326:
2323:
2303:
2300:
2297:
2294:
2264:
2260:
2243:
2240:
2213:
2209:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2134:
2131:
2128:
2108:
2105:
2102:
2099:
2079:
2076:
2073:
2070:
2066:
2063:
2042:
2039:
2035:
2032:
2011:
1990:
1987:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1944:
1941:
1937:
1933:
1930:
1926:
1906:
1901:
1897:
1868:
1865:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1818:
1814:
1797:
1794:
1790:Sacks property
1786:
1785:
1717:
1714:
1713:
1712:
1701:
1679:
1651:random algebra
1649:Main article:
1646:
1645:Random forcing
1643:
1627:
1624:
1623:
1622:
1569:
1562:Easton product
1558:
1509:and such that
1468:
1458:
1453:
1446:
1437:
1430:
1420:
1413:
1406:
1399:
1365:
1362:
1275:
1274:Prikry forcing
1272:
1247:
1243:
1227:
1226:
1222:
1204:
1200:
1175:
1161:
1158:
1153:
1149:
1145:
1142:
1123:
1115:
1111:
1108:
1104:Adrian Mathias
1100:
1099:
1023:
1022:
1001:An element of
997:
994:
985:
982:
974:
973:
953:on subsets of
940:
931:
914:Main article:
911:
908:
904:Laver property
861:
860:
818:
766:
737:
736:
717:
714:
670:Main article:
667:
664:
649:
646:
642:
611:
607:
583:
578:
574:
535:
530:
526:
513:
510:
396:
393:
384:
381:
376:
369:
342:
327:Main article:
324:
321:
312:
311:Amoeba forcing
309:
308:
307:
280:
181:
160:there is some
136:
117:satisfies the
110:
107:
106:
105:
91:
85:
79:
71:
68:
9:
6:
4:
3:
2:
3677:
3666:
3663:
3662:
3660:
3651:
3650:
3645:
3644:
3635:
3631:
3627:
3621:
3617:
3613:
3609:
3606:
3600:
3596:
3595:
3590:
3586:
3583:
3577:
3573:
3569:
3565:
3561:
3560:
3551:
3542:
3538:
3530:
3511:
3507:
3495:
3492:
3485:-generic and
3448:
3444:
3423:
3420:
3417:
3396:
3393:
3389:
3386:
3364:
3360:
3339:
3318:
3315:
3289:
3286:
3265:
3262:
3219:
3199:
3196:
3193:
3170:
3167:
3164:
3158:
3155:
3150:
3146:
3139:
3136:
3133:
3110:
3107:
3104:
3101:
3098:
3074:
3071:
3067:
3064:
3044:
3041:
3038:
3018:
3015:
3012:
2992:
2989:
2986:
2966:
2943:
2890:
2887:
2878:
2838:
2830:
2820:
2818:
2813:
2811:
2807:
2803:
2799:
2795:
2791:
2787:
2781:
2777:
2767:
2757:
2755:
2733:
2730:
2727:
2721:
2718:
2715:
2703:
2694:
2690:
2664:
2658:
2650:
2634:
2631:
2628:
2608:
2605:
2602:
2594:
2575:
2572:
2569:
2543:
2540:
2537:
2514:
2511:
2508:
2488:
2485:
2482:
2462:
2459:
2453:
2450:
2447:
2424:
2421:
2418:
2410:
2392:
2388:
2379:
2369:
2353:
2340:
2324:
2321:
2298:
2292:
2284:
2280:
2262:
2258:
2249:
2239:
2237:
2233:
2229:
2228:Ronald Jensen
2211:
2198:
2194:
2172:
2169:
2163:
2160:
2157:
2145:
2136:
2132:
2126:
2103:
2097:
2077:
2074:
2071:
2068:
2064:
2061:
2040:
2037:
2033:
2030:
2009:
1988:
1985:
1961:
1958:
1955:
1949:
1942:
1939:
1935:
1931:
1928:
1917:, ordered by
1899:
1895:
1886:
1882:
1866:
1863:
1856:
1853:
1850:
1841:
1838:
1816:
1812:
1803:
1793:
1791:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1748:so that both
1747:
1743:
1739:
1735:
1731:
1723:
1720:
1719:
1716:Sacks forcing
1709:
1704:
1700:
1695:
1691:
1687:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1655:
1654:
1652:
1642:
1640:
1636:
1631:
1626:Radin forcing
1618:
1611:
1607:
1603:
1599:
1594:
1590:
1586:
1581:
1577:
1572:
1568:
1563:
1559:
1556:
1550:
1546:
1542:
1538:
1532:
1528:
1523:
1517:
1513:
1506:
1502:
1498:
1494:
1489:
1485:
1480:
1476:
1471:
1467:
1462:
1459:
1452:
1445:
1436:
1429:
1419:
1412:
1405:
1398:
1391:
1387:
1382:
1378:
1374:
1371:
1370:
1369:
1361:
1358:
1354:
1349:
1345:
1341:
1337:
1333:
1327:
1323:
1315:
1311:
1305:
1301:
1297:
1291:
1287:
1281:
1271:
1269:
1265:
1261:
1245:
1232:
1220:
1202:
1189:
1185:
1181:
1159:
1156:
1151:
1147:
1143:
1140:
1132:
1129:
1128:
1127:
1121:
1118:to ω without
1110:Namba forcing
1107:
1105:
1096:
1092:
1087:
1083:
1079:
1075:
1071:
1065:
1061:
1057:
1053:
1046:
1042:
1034:
1030:
1025:
1024:
1020:
1016:
1012:
1008:
1004:
1000:
999:
993:
991:
981:
979:
971:
962:(α, ξ) < α
961:
952:
948:
944:
941:
938:
935:
932:
929:
926:
923:
922:
921:
917:
907:
905:
900:
898:
894:
888:
885:) : p ∈
884:
880:
872:
866:
856:
851:
847:
843:
838:
834:
828:
824:
819:
816:
812:
807:
803:
797:
793:
787:
783:
779:
775:
770:
767:
764:
760:
756:
752:
749:
748:
747:
745:
742:
734:
731:
730:
729:
727:
723:
716:Laver forcing
713:
711:
707:
703:
699:
695:
691:
687:
683:
679:
673:
663:
647:
644:
640:
631:
628:
624:
609:
605:
581:
576:
563:
559:
555:
554:Carl Jockusch
551:
533:
528:
516:Forcing with
509:
507:
503:
497:
493:
489:
485:
480:
476:
472:
468:
464:
460:
456:
450:
446:
440:
436:
432:
428:
424:
418:
414:
408:
404:
402:
392:
390:
380:
373:
366:
363:
359:
353:
349:
340:
336:
330:
329:Cohen forcing
323:Cohen forcing
320:
318:
305:
301:
297:
293:
289:
285:
281:
277:
273:
267:
263:
257:
253:
247:
243:
237:
233:
227:
223:
217:
213:
207:
203:
199:such that if
198:
194:
190:
186:
182:
178:
174:
168:
164:
158:
154:
150:if for every
149:
145:
141:
137:
134:
130:
126:
122:
121:
116:
113:
112:
103:
99:
95:
92:
89:
86:
83:
80:
77:
74:
73:
67:
65:
61:
57:
53:
49:
46:
42:
38:
34:
30:
19:
3648:
3615:
3597:, Elsevier,
3592:
3567:
3564:Jech, Thomas
3550:
3541:
3254:, and hence
2829:Petr Vopěnka
2826:
2814:
2809:
2805:
2801:
2797:
2793:
2789:
2785:
2779:
2775:
2763:
2753:
2648:
2595:then either
2592:
2408:
2377:
2375:
2338:
2282:
2278:
2247:
2245:
2196:
2192:
2002:end-extends
1884:
1880:
1801:
1799:
1787:
1781:
1773:
1769:
1765:
1761:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1721:
1707:
1702:
1698:
1693:
1689:
1685:
1680:
1676:
1672:
1668:
1664:
1660:
1656:
1639:supercompact
1632:
1629:
1616:
1609:
1605:
1601:
1597:
1592:
1588:
1584:
1579:
1575:
1570:
1566:
1561:
1554:
1548:
1544:
1540:
1536:
1530:
1526:
1521:
1515:
1511:
1504:
1500:
1496:
1492:
1487:
1483:
1478:
1474:
1469:
1465:
1460:
1450:
1443:
1434:
1427:
1417:
1410:
1403:
1396:
1389:
1385:
1380:
1376:
1372:
1367:
1356:
1352:
1347:
1343:
1339:
1335:
1331:
1325:
1321:
1313:
1309:
1303:
1299:
1295:
1289:
1285:
1279:
1277:
1262:
1230:
1228:
1218:
1187:
1183:
1179:
1130:
1113:
1101:
1094:
1090:
1085:
1081:
1077:
1073:
1069:
1063:
1059:
1055:
1051:
1044:
1040:
1032:
1028:
1018:
1014:
1010:
1006:
1002:
987:
975:
969:
959:
950:
946:
942:
936:
933:
927:
924:
919:
901:
896:
892:
886:
882:
878:
870:
864:
862:
854:
849:
845:
841:
836:
832:
826:
822:
814:
810:
805:
801:
795:
791:
785:
781:
777:
773:
768:
762:
758:
754:
750:
743:
740:
738:
732:
719:
675:
561:
550:Robert Soare
515:
505:
501:
495:
491:
487:
483:
478:
474:
470:
466:
462:
458:
454:
448:
444:
438:
434:
430:
426:
422:
416:
412:
406:
405:
400:
398:
386:
374:
367:
361:
357:
351:
347:
338:
332:
317:amoeba order
314:
303:
299:
295:
291:
287:
283:
275:
271:
265:
261:
255:
251:
245:
241:
235:
231:
225:
221:
215:
211:
205:
201:
196:
192:
188:
184:
176:
172:
166:
162:
156:
152:
147:
143:
139:
132:
128:
124:
118:
114:
101:
97:
93:
87:
81:
75:
63:
59:
55:
54:. The poset
51:
47:
40:
32:
26:
3352:. Defining
978:Azriel Levy
891:, called a
753:is a tree:
702:Baumgartner
596:subsets of
477:but not of
391:on ω.
389:ultrafilter
109:Definitions
50:to a model
3634:1262.03001
3616:Set theory
3533:References
3410:such that
3186:, for all
1760:.) A tree
1744:extending
1635:measurable
1190:which has
1120:collapsing
964:for every
955:λ × κ
893:Laver-real
789:such that
741:Laver tree
682:Tennenbaum
627:computable
335:Paul Cohen
290:is called
146:is called
37:set theory
3496:∈
3421:∈
3390:∈
3340:α
3290:∈
3200:α
3197:⊆
3165:γ
3159:φ
3156:⊨
3151:β
3143:⇔
3137:∈
3111:φ
3105:γ
3099:β
3068:∈
3042:⊆
3016:≤
2993:α
2990:⊆
2967:α
2944:α
2731:⋃
2728:∈
2725:⟩
2722:β
2716:α
2713:⟨
2704:β
2701:∃
2695::
2691:α
2635:α
2629:δ
2609:γ
2603:β
2579:⟩
2576:δ
2570:γ
2567:⟨
2547:⟩
2544:β
2538:α
2535:⟨
2512:∈
2509:α
2489:β
2486:≤
2483:α
2460:∈
2457:⟩
2454:β
2448:α
2445:⟨
2422:∈
2389:ω
2350:ℵ
2322:⋃
2259:ω
2208:ℵ
2170:∈
2167:⟩
2158:σ
2155:⟨
2143:∃
2137::
2133:σ
2127:⋃
2075:∪
2072:σ
2069:⊆
2062:σ
2038:⊆
2010:σ
1986:σ
1965:⟩
1956:σ
1953:⟨
1950:≤
1947:⟩
1929:σ
1925:⟨
1896:ω
1867:σ
1864::
1860:⟩
1851:σ
1848:⟨
1813:ω
1756:1 are in
1242:ℵ
1199:ℵ
1160:ω
1148:ω
1144:⊆
648:ω
610:ω
573:Π
525:Π
469:, and if
282:A subset
229:, and if
138:A subset
3659:Category
3614:(2011),
3591:(1980),
3566:(2003),
3397:′
3319:′
3266:′
3075:′
2979:, where
2891:′
2808:extends
2784:, where
2234:and the
2065:′
2034:′
1989:′
1943:′
1932:′
1583:, where
1553:for all
1093:∪
1058:) < (
809:for all
630:subtrees
560:. Here
500:for all
70:Notation
3436:, then
2407:we set
2277:we set
1831:we set
1696:, then
1641:, etc.
1619:(α) ≠ 1
1388:×
1264:Magidor
990:Magidor
678:Solovay
490:) >
292:generic
29:forcing
3632:
3622:
3601:
3578:
3126:where
2770:{0, 1}
2090:. In
1752:0 and
1726:{0, 1}
1346:, and
1294:where
1268:Shelah
1084:, and
966:(α, ξ)
710:Shelah
706:Shelah
690:Easton
421:where
185:filter
2475:then
2285:. In
1595:with
1518:) = 1
1490:with
1409:) ≤ (
1375:: If
840:then
722:Laver
698:Laver
481:then
350:<
294:over
259:with
219:then
204:<
170:with
148:dense
100:over
45:poset
43:of a
3620:ISBN
3599:ISBN
3576:ISBN
3031:iff
2632:<
2606:<
2559:and
2501:and
2437:and
2376:For
2341:and
2246:For
2053:and
2022:and
1977:iff
1883:and
1800:For
1604:) ∈
1560:The
1543:) ≤
1499:) ∈
1441:and
1379:and
1266:and
1157:<
873:, ≤)
852:for
830:and
780:) =
680:and
645:<
552:and
269:and
239:and
209:and
131:and
3630:Zbl
3500:HOD
3472:HOD
3463:is
3294:HOD
2959:of
2863:HOD
2812:.
2804:if
2647:.
2621:or
1768:if
1736:of
1591:on
1534:if
1486:on
1425:if
1330:if
1182:in
1068:if
863:If
857:∈ ω
848:in
820:If
813:in
799:or
632:of
504:in
453:if
355:if
302:in
286:of
195:of
187:on
142:of
35:of
3661::
3628:,
3574:,
3529:.
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2913:OD
2778:,
2238:.
2199:.
1792:.
1706:∈
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1402:,
1355:∪
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1324:,
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1288:,
1126:.
1106:.
1076:,
1066:))
1062:,
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1050:((
1043:,
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980:.
906:.
899:.
846:tn
835:≤
825:∈
804:≤
794:≤
784:∈
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688:.
508:.
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447:,
437:,
415:,
365:.
360:⊇
337:)
274:≤
264:≤
254:∈
244:∈
234:∈
224:∈
214:∈
183:A
175:≤
165:∈
155:∈
3517:]
3512:A
3508:G
3504:[
3493:A
3449:A
3445:G
3424:p
3418:A
3394:P
3387:p
3365:A
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3220:p
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3171:x
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3096:(
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2710:(
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2688:{
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1962:C
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1936:,
1905:}
1900:1
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