## On what I do not understand (and have something to say), model theory

Venue: | Mathematica Japonica, submitted. [Sh:702]; math.LO/9910158 |

Citations: | 23 - 8 self |

### BibTeX

@INPROCEEDINGS{Shelah_onwhat,

author = {Saharon Shelah},

title = {On what I do not understand (and have something to say), model theory},

booktitle = {Mathematica Japonica, submitted. [Sh:702]; math.LO/9910158},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (“see... ” means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall ’97 and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Ros̷lanowski for many helpful comments. (666) revision:2001-11-12 modified:2003-11-18

### Citations

177 | Classification theory and the number of nonisomorphic models - Shelah - 1978 |

115 |
The complexity of the pigeonhole principle
- Ajtai
- 1994
(Show Context)
Citation Context ...e for fragments of PA (with bounded induction, without (∀x)(2 x exists) (possibly with (∀x)(x [log(x)] exists)), they speak about pigeonhole principles when we add a new set as a predicate (see Ajtai =-=[5]-=-), this seems to me parallel to the Frankel–Mostowski method. I have thought about it lately because of [124]. Concerning 9.3, let us consider the following thought. In ZFC + V = L, every element has ... |

75 |
The Axiom of Determinacy, Forcing Axioms and the Non-stationary Ideal, volume 1 of Logic and its Applications. de Gruyter
- Woodin
(Show Context)
Citation Context ...ith the case of κ = ℵ1 and the consistency strength ZFC (see Ros̷lanowski and Shelah [88, §3]); the theory of determinacy is applicable to the case κ = ℵ1 and maximal consistency strength (see Woodin =-=[182]-=-), and [115] intends to deal with the case of “ZFC + κ is strongly inaccessible” (and no further consistency strength assumptions).s34 SAHARON SHELAH Question 4.29. Is there a sweet forcing notion (se... |

61 |
Combinatorial Set Theory: Partition Relations for Cardinals
- Erdös, Hajnal, et al.
- 1984
(Show Context)
Citation Context ...igid, or 1-to-1 rigid) R–module of size ≥ λ, then there are such R-modules in arbitrarily large cardinals? (I.e. Hanf numbers). 8. Partitions and colourings Remember (see Erdős, Hajnal, Maté and Rado =-=[33]-=-) Definition 8.1. 1. λ → (α) n κ means: for every colouring c : [λ] n −→ κ there is a set X ⊆ λ of order type α such that c ↾ [X] n is constant. 2. λ → [α] n κ means: for every colouring c : [λ] n −→ ... |

54 |
Primitive recursive bounds for van der Waerden numbers
- Shelah
- 1988
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Citation Context ... ℓ ∗ } ∪ {xℓ ∗}) = F({xℓ, yℓ : ℓ �= ℓ ∗ } ∪ {yℓ ∗}). Again the main question for me is: Does f ∗ (n, c) grow as a fixed iterated exponentiation? (This is connected to the van der Waerden theorem, see =-=[143]-=-). On the Ramsey Theory see Graham, Rothschild and Spencer [47]. Definition 8.20. 1. For a group G and a subset A of G, and a group H let H → (G) A σ mean: if d is a function with domain H and range o... |

50 |
Almost Free Modules: Set-Theoretic Methods
- Eklof, Mekler
- 2002
(Show Context)
Citation Context ... ∩ {xβ : β < λ} ⊆ {xβ : β ≤ α}. Now, Question 6.19. Are the cardinal bounds in 6.14 — 6.17 best possible? 7. A taste of Algebra I have much interest in Abelian groups, but better see Eklof and Mekler =-=[31]-=-. Thomas prefers to deal just with short elegant proofs of short elegant problems (for me the second demand suffices). So he was rightly happy when proving that for any infinite group G with no center... |

50 |
Cardinal arithmetic, volume 29 of Oxford Logic Guides. The Clarendon Press Oxford
- Shelah
- 1994
(Show Context)
Citation Context ...66 Based on lectures in the Rutgers Seminar, Fall 1997 are: §1, §2, §5, §7, §8. 1s2 SAHARON SHELAH 1. Cardinal problems and pcf Here, we deal with cardinal arithmetic as I understand it (see [150] or =-=[154]-=-), maybe better called cofinality arithmetic (see definitions below). What should be our questions? Wrong questions usually have no interesting answers or none at all. Probably the most popular 1 ques... |

49 |
Set Theory: On the Structure of the Real Line. A K Peters
- Bartoszyński, Judah
- 1995
(Show Context)
Citation Context ...ON WHAT I DO NOT UNDERSTAND 19 Discussion 3.3. Concerning 2.9, I had not really considered it (except when Judah spoke to me about it) but just before the lecture, Bartoszyński reminded me of it (see =-=[6]-=-). Now, “the” proof of CON(Borel conjecture) is by CS iteration of Laver forcing (see Laver [73]), whereas the consistency proof of the dual is adding many Cohen reals (see Carlson [18]). So in a (hop... |

44 |
Classification theory for nonelementary classes. I. the number of uncountable models of ψ ∈ Lω1ω part A
- Shelah
- 1983
(Show Context)
Citation Context ...: n<ω B(Iα). In the positive u ⊆ A, u �= A}, and a sequence of liftings ¯ f = 〈fs : s ∈ P− (n)〉 satisfying a reasonable induction hypothesis, � fs can be extended to a lifting of � Is,α (as s e.g. in =-=[133]-=-, Sageev and Shelah [90]). For the negative direction we may think of using a partition theorem. ∗ ∗ ∗ For a long time I have been interested in compactness in singular cardinals; i.e., whether if som... |

44 |
Can you take Solovay’s inaccessible away
- Shelah
- 1984
(Show Context)
Citation Context ...prove independence except forcing? As mathematicians do not report their failures, not much is said in the literature. I do not mean results that simply follow from the consistency strength, e.g., by =-=[146]-=- in some forcing extensions of L by a forcing notion not collapsing cardinals (in fact satisfying the c.c.c.), PB ≡“ every projective set of reals has the Baire property ”, whereas the consistency str... |

43 |
and improper forcing
- Proper
- 1998
(Show Context)
Citation Context ... a set Y from W of cardinality < κ including X which is the universe of an elementary submodel of M). 2. The quest for the test: on the theory of Iterated Forcing for the continuum On the subject see =-=[164]-=-, and recent papers, too, but this section is hampered by some works in progress. The issue is: i<κs12 SAHARON SHELAH Problem 2.1. (a) Assuming we know something about each iterand Q ˜ i, what can we ... |

36 |
On the consistency of Borel 's conjecture
- Laver
(Show Context)
Citation Context ...except when Judah spoke to me about it) but just before the lecture, Bartoszyński reminded me of it (see [6]). Now, “the” proof of CON(Borel conjecture) is by CS iteration of Laver forcing (see Laver =-=[73]-=-), whereas the consistency proof of the dual is adding many Cohen reals (see Carlson [18]). So in a (hopeful) iteration proving consistency we have two kinds of assignments. We are given, say in stage... |

36 |
groups may not be free even assuming CH
- Shelah, Whitehead
- 1980
(Show Context)
Citation Context ...low (5.2, 5.3, 5.4) is a difference between asking about Sλ θ = {δ < λ : cf(δ) = θ} and asking about a stationary S ⊆ Sλ θ such that Sλ θ \S is stationary too; a difference of which I became aware in =-=[126]-=- (e.g. ♦S1 & ¬♦S2 is possible for disjoint stationary subsets of ω1) after much agony. Question 5.2 (GCH). If µ is singular, do we have ♦ µ S + ? (Those are the only cf(µ) cases left.) Similarly for i... |

31 | Cardinal arithmetic - Shelah - 1994 |

28 |
A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals
- Shelah
- 1975
(Show Context)
Citation Context ...mentary submodels of (H(χ), ∈) such that x ∈ M0 and ¯ M ↾ (i + 1) ∈ Mi+1, then for some club C of µ + , i ∈ C ⇒ Mi ∩ B ∗ ∈ Y . On the filters see Kueker [70], and [123, §3]. The theorem was proved in =-=[123]-=- but with two extra axioms, however it included the full case of varieties (i.e., including the non-Schreier ones). Later, the author eliminated those two extra axioms: Ax V in Ben David [13], and Ax ... |

27 |
On the consistency of some partition theorems for continuous colorings and the structure of ℵ1-dense real order type, Annals of Pure and Applied Logic 29
- Abraham, Rubin, et al.
- 1985
(Show Context)
Citation Context ..., J are θ–far if there is no linear order of cardinality θ embedded into both. If θ is omitted, we mean min{|I|, |J|}. On OCA ′ (i.e., OCAℵ1,ω, see the definition below) see Abraham, Rubin and Shelah =-=[2]-=-, continued for OCA ′′ in Todorčević [178], Veličković [180]; on a parallel for subsets of the plain which follows from MA, see Steprāns and Watson [176]. Question 2.23. 1. Is OCA ′ ℵ2 consistent? Is ... |

27 |
The Generalized Continuum Hypothesis Revisited ([Sh460
- Shelah
(Show Context)
Citation Context ... when they say something at all, but Problem 1.1 is always meaningful. To present what I think are central problems, we can start from what I called the solution of the “Hilbert’s first problem”, see =-=[119]-=- (though without being seconded). Theorem 1.5. For λ ≥ �ω, there are κ < �ω and P ⊆ [λ] <�ω , |P| = λ such that every A ∈ [λ] <�ω is equal to the union of < κ members of P. 1 that is, most people who ... |

26 | Categoricity in abstract elementary classes: going up inductively - Shelah |

21 |
Higher Souslin trees and the generalized continuum hypothesis
- Gregory
- 1976
(Show Context)
Citation Context ... + , λ ++ (λ regular). How can this help? Assume that there is no λ ++ –Souslin tree and GCH holds. It follows is40 SAHARON SHELAH that every stationary S ⊆ S λ++ ≤λ reflects in Sλ++ λ + (see Gregory =-=[48]-=-), moreover it is enough to assume just that there is no (< λ + )–complete Souslin tree, by Kojman and Shelah [68]. Hence (⊗) there is S ∗ ⊆ λ + such that: (a) (square on S ∗ ) ¯C = 〈Cα : α ∈ S ∗ 〉, C... |

20 | Reflecting stationary sets and successors of singular cardinals - Shelah - 1991 |

20 |
Viva la difference I: Nonisomorphism of ultrapowers of countable models
- Shelah
- 1992
(Show Context)
Citation Context ...more to be said and done. Of course, we can now carry out generalizations of various independence results on cardinal invariants, e.g. between variants of (f, g)–bounding and slaloms (= corsets), see =-=[151]-=-, [164, Ch.VI,§2], Goldstern and Shelah [46], Ros̷lanowski and Shelah [87]. We may consider in the tree version for every limit δ ∈ S (S ⊆ κ is a stationary set) to omit tops for one or just ≤ |δ| bra... |

19 |
Proper forcing
- Shelah
- 1982
(Show Context)
Citation Context ...by a not so nice example that generally you cannot just omit the (< ω1)–properness demand and promise an iteration theorem covering them. The problem concerning that forcing was resolved (promised in =-=[138]-=-, carried out in a different way in lectures in MSRI ’89 = [164, Ch.XVIII,§2]), but resulted in a dichotomy: we can get by forcing CON(ZFC+CH+SH) and we can get by forcing CON(ZFC + CH + no club guess... |

16 |
On successors of singular cardinals
- Shelah
- 1978
(Show Context)
Citation Context ... Can we get something parallel when cf(µ) = ℵ0? Question 5.7. Can we prove that for some strong limit singular cardinal µ and a regular cardinal θ < µ we have ♣S, where S = S µ+ θ ? ∗ ∗ ∗ On I[λ] see =-=[128]-=-, [141], [152]. We know that e.g. is consistent with GCH, but {δ < ℵω+1 : cf(δ) = ℵ1} /∈ I[ℵω+1] Problem 5.8. 1. Can {δ < ℵω+1 : cf(δ) = ℵ2} /∈ I[ℵω+1]? 2. Can {δ < (2 ℵ0 ) +ω+1 : cf(δ) = (2 ℵ0 ) + } ... |

16 |
Proper forcing, volume 940
- Shelah
- 1982
(Show Context)
Citation Context ...see Abraham [1], Cummings and Foreman [22]). A relevant problem is 6.4. ∗ ∗ ∗ I have found partition theorems on trees with ω levels very useful and interesting (see Rubin and Shelah [89], and [134], =-=[132]-=-, [164, X, XI, XV,2.6]). In [145, 13,p.1453] and [127, Ch.VIII,§1] trying to prove a theorem on the number of nonisomorphic models of a pseudo elementary class we arrived at the following problem [wit... |

16 |
Partition Problems in Topology, volume 84 of Contemporary Mathematics
- Todorčević
- 1989
(Show Context)
Citation Context ...r of cardinality θ embedded into both. If θ is omitted, we mean min{|I|, |J|}. On OCA ′ (i.e., OCAℵ1,ω, see the definition below) see Abraham, Rubin and Shelah [2], continued for OCA ′′ in Todorčević =-=[178]-=-, Veličković [180]; on a parallel for subsets of the plain which follows from MA, see Steprāns and Watson [176]. Question 2.23. 1. Is OCA ′ ℵ2 consistent? Is OCA′′ ℵ2 consistent? 2. The parallel probl... |

15 |
All ℵ1-dense sets of reals can be isomorphic
- Baumgartner
- 1973
(Show Context)
Citation Context ... ⊆ κ codes H(χ), κ large and try to collapse it to ω2. Note that by [105], if CH + Dω1 is ℵ2–saturated, then essentially we have the weak diamond for three colours (or any finite number). Baumgartner =-=[11]-=- asked ∗ ∗ ∗ Question 2.20. Is it consistent that 2 ℵ0 > ℵ2 and any two ℵ2–dense subsets of R of cardinality ℵ2 (that is, any interval has ℵ2 points) are isomorphic (as linear orders). I think it is m... |

15 |
The Covering Lemma for K
- Dodd, Jensen
- 1982
(Show Context)
Citation Context ...here may well be some “global phenomena”. Also there may be special behaviour near min{λ : for some A ⊆ λ, there is no indiscernible class for K[A]}, as above it the covering theorem (Dodd and Jensen =-=[26]-=-) shows that cardinal arithmetic is trivial. On the other hand, on the behaviour below it, see [154, Ch.V]. An extreme case of our non-understanding concerning global behavior is: Question 1.22. Is it... |

15 |
Rigid boolean algebras
- Bekkali, Bonnet
(Show Context)
Citation Context ... Boolean Algebras is proved: Theorem 1.6. If B is a c.c.c. Boolean algebra and µ = µ �ω ≤ |B| ≤ 2 µ , then B is µ-linked, i.e., B\{0} is the union of µ sets of pairwise compatible elements. (See also =-=[129]-=-, [131] and Hajnal, Juhász and Szentmiklossy [53].) We also have the following application: Theorem 1.7 (See [155]). If X is a topological space (not necessarily T2) with λ points, µ ≤ λ < 2 µ and > λ... |

14 |
Real-valued measurable cardinals
- Solovay
- 1971
(Show Context)
Citation Context ...bra which up to isomorphism is equivalent to) the Cohen forcing or random real forcing, an old question which Fremlin promoted (see [37]), which comes from asking: can the classical result of Solovay =-=[175]-=- (saying that consistently 2 ℵ0 is real valued measurable, now the Maharam type there was large) be improved to get small Maharam type. But then [42] turns to: Problem 4.13. 1. Prove that P(κ)/I canno... |

13 |
and Saharon Shelah. There may be simple Pℵ1- and Pℵ2-points and the RudinKeisler ordering may be downward directed
- Blass
- 1987
(Show Context)
Citation Context ...ing notions as in Ros̷lanowski and Shelah [85], [87], which proved easily checkedsON WHAT I DO NOT UNDERSTAND 21 sufficient conditions for what we desire (so in the “neighborhood” of Blass and Shelah =-=[14]-=-). But what should be the norm? ∗ ∗ ∗ Discussion 3.7. Concerning p < t, I have made quite a few failed tries. Some try to use long iterations (≥ ℵω+1) or a new support. But also I thought that Blass a... |

13 |
Strong measure zero and strongly meager sets
- Carlson
- 1993
(Show Context)
Citation Context ...d me of it (see [6]). Now, “the” proof of CON(Borel conjecture) is by CS iteration of Laver forcing (see Laver [73]), whereas the consistency proof of the dual is adding many Cohen reals (see Carlson =-=[18]-=-). So in a (hopeful) iteration proving consistency we have two kinds of assignments. We are given, say in stage α, in VPα a set A = {ηi : i < ω1} ⊆ ω2, and we should make it not of strong measure zero... |

13 |
Cardinal Invariants of Boolean Algebras, volume 142
- Monk
- 1996
(Show Context)
Citation Context ...are i < j such that If |I| = µ then we omit µ. ti,ℓ < tj,ℓ ⇔ ℓ ∈ w. Question 5.12. Is there an entangled linear order of cardinality λ + , where λ = λ ℵ0 ? A “yes” answer will solve a problem of Monk =-=[80]-=- on the spread of ultraproducts of Boolean Algebras; see [162]. With the help of pcf we can build entangled linear orders in λ + for many λ which means: provably for a proper class of λ’s. The interes... |

13 |
Correction to: Adding a random or a Cohen real: topological consequences and the effect on Martin’s axiom, Fund
- Roitman
- 1988
(Show Context)
Citation Context ...force by some P, which makes d = ℵ1, but P is understood well enough and we can show that a is still large (just as adding a Cohen real to a model of MA preserves some consequences of MA (see Roitman =-=[81]-=-, Judah and Shelah [58]]). So clearly FS iteration will not do. I think that a more interesting way is to consider, assuming CH, Kω1 = {( ¯ P, ¯r ˜ ) : ¯ P = 〈Pi : i < ω1〉 is <◦–increasing, |Pi| ≤ ℵ1,... |

13 |
Analytic ideals and their applications
- Solecki
(Show Context)
Citation Context ... as a forcing is ω ω–bounding, a measure algebra. Another way to express the thought that Cohen and random are special was Kunen’s conjecture, see Kunen [71], Kechris and Solecki [65], Solecki [174], =-=[173]-=- and Ros̷lanowski and Shelah [86]. It is natural to investigate the partial orders from 4.8. So,sON WHAT I DO NOT UNDERSTAND 29 Problem 4.9. Investigate the quasi order ≤ ∗ (and its variants) for Q wh... |

12 |
Strong partition relations below the power set: Consistency, was Sierpinski right
- Shelah
- 1991
(Show Context)
Citation Context ... (just as, e.g., the Erdös-Rado theorem helps in ZFC results). Still it seems to me worthwhile to know. Question 8.13. 1. Can we put together the results of, e.g., [121], Shelah and Stanley [167] and =-=[149]-=-? Assume that κ = κ <κ < λ, λ is, e.g., strongly inaccessible large enough. Can we find a (< κ)–complete, κ + –c.c. forcing notion P such that in V P : (a) for σ < κ and µ < λ we can find µ ′ , λ ′ su... |

12 | Cardinal arithmetic for skeptics
- Shelah
- 1992
(Show Context)
Citation Context ...ication 666 Based on lectures in the Rutgers Seminar, Fall 1997 are: §1, §2, §5, §7, §8. 1s2 SAHARON SHELAH 1. Cardinal problems and pcf Here, we deal with cardinal arithmetic as I understand it (see =-=[150]-=- or [154]), maybe better called cofinality arithmetic (see definitions below). What should be our questions? Wrong questions usually have no interesting answers or none at all. Probably the most popul... |

12 |
Saturated filters at successors of singulars, weak reflection and yet another weak club principle
- Dzamonja, Shelah
(Show Context)
Citation Context ...ingular) 2 µ = µ + + □µ + ⇒ ♦ S µ+ cf(µ) when cf(µ) > ℵ0 (see [147, 3.2,p.1030]). So we need large cardinals (hardly surprising for successors of singular cardinals). See more in Dˇzamonja and Shelah =-=[31]-=-).48 SAHARON SHELAH (666) revision:2001-11-12 modified:2003-11-18 Probably it is wiser to try to force this for “large” µ. Changing the cofinality of a supercompact cardinal µ to ℵ0, where “µ is prep... |

11 |
and Saharon Shelah, Cardinal invariants above the continuum, Annals of Pure and Applied Logic 75
- Cummings
- 1995
(Show Context)
Citation Context ...ars. Problem 5.15. Prove the consistency of: for every λ (or regular λ) a suitable forcing axiom holds. Relevant is “GCH fails everywhere” (see Foreman and Woodin [35]). Now Cummings and Shelah [23], =-=[24]-=- is a modest try and 1.22 is relevant. Specific well known targets are Problem 5.16. Is GSH consistent? (GSH is the generalized Souslin hypothesis: for every regular uncountable λ there is no λ-Sousli... |

11 |
All uncountable cardinals can be singular
- Gitik
- 1980
(Show Context)
Citation Context ...the existence of a proper class of regular cardinals? (b) Does DCκ for κ large enough imply that for a class of λ, P(λ) is not the union of < λ sets, each of cardinality ≤ λ? See more in [163]. Gitik =-=[40]-=- had proved CON � (∀δ)(cf(δ) ≤ ℵ0) � relative to suitable large cardinals. Woodin asked if CON(DCℵ0 + (∀δ)(cf(δ) ≤ ℵ1)). Specker asked if, consistently, for every λ, for some 〈An � : n < ω〉 we have P(... |

11 |
Decomposing uncountable squares to countably many chains
- Shelah
- 1976
(Show Context)
Citation Context ...hich none of them embeds is necessarily a Specker order (= take an Aronszajn tree, order it lexicographically). You may ask: Can there be a “minimal” order among those? But there cannot. It is known (=-=[124]-=-, answering a question of Countryman) that there is a Specker order L such that the product L × L (with the product order) is the union of countably many chains (comes from a very special Aronszajn tr... |

11 | special are Cohen and random forcings i.e. Boolean algebras of the family of subsets of reals modulo meagre or null
- How
- 1994
(Show Context)
Citation Context ...ts. I feel the former version is more interesting. For me this translates to Problem 4.5. Among Souslin c.c.c. forcing notions, are Cohen forcing and random forcing special? Some progress was made in =-=[156]-=-. Theorem 4.6. If a Souslin c.c.c. forcing notion Q adds η ˜ ∈ ω ω not dominated by any old ν ∈ ω ω, then forcing with Q adds a Cohen real. (The “Souslin” is needed for enough absoluteness, so with th... |

11 | Analytic ideals
- Solecki
- 1996
(Show Context)
Citation Context ...a which as a forcing is ω ω–bounding, a measure algebra. Another way to express the thought that Cohen and random are special was Kunen’s conjecture, see Kunen [71], Kechris and Solecki [65], Solecki =-=[174]-=-, [173] and Ros̷lanowski and Shelah [86]. It is natural to investigate the partial orders from 4.8. So,sON WHAT I DO NOT UNDERSTAND 29 Problem 4.9. Investigate the quasi order ≤ ∗ (and its variants) f... |

10 |
and Saharon Shelah, A weak version of ♦ which follows from 2
- Devlin
- 1978
(Show Context)
Citation Context ...t would be easy to derive an axiom; some years later this materialized as Abraham, Devlin and Shelah [5], but reality is not as nice as dreams. One obstacle is the weak diamond, see Devlin and Shelah =-=[25]-=-, more in [132, Ch.XIV,§1], [164, AP,§1], [105]. For a time the iteration theorem in [132, Ch.V,§5,§7,Ch.VIII,§4] seemed satisfactory to me. [There we use two demands. The first was D–completeness (th... |

10 |
The generalized continuum hypothesis can fail everywhere
- Foreman, Woodin
- 1991
(Show Context)
Citation Context ...control in the neighborhood of singulars. Problem 5.15. Prove the consistency of: for every λ (or regular λ) a suitable forcing axiom holds. Relevant is “GCH fails everywhere” (see Foreman and Woodin =-=[35]-=-). Now Cummings and Shelah [23], [24] is a modest try and 1.22 is relevant. Specific well known targets are Problem 5.16. Is GSH consistent? (GSH is the generalized Souslin hypothesis: for every regul... |

10 |
Infinite Abelian Groups, Volume I
- Fuchs
- 1970
(Show Context)
Citation Context ..., Mekler, Shelah and Spinas [78].) ∗ ∗ ∗ There are cases of strong dichotomy: if ≥ λ then ≥ 2 λ , related to groups (see [142], Grossberg and Shelah [49], [50], and [118]; on Abelian groups see Fuchs =-=[38]-=-). Question 7.10. [V = L] If λ > cf(λ) > ℵ0, G is a torsion free Abelian group of cardinality λ, can λ = νp(Ext(G, Z))?s50 SAHARON SHELAH The cardinal νp(Ext(G, Z)) is the dimension of {x ∈ Ext(G, Z) ... |

10 |
and Saharon Shelah. Many simple cardinal invariants
- Goldstern
- 1993
(Show Context)
Citation Context ...ow carry out generalizations of various independence results on cardinal invariants, e.g. between variants of (f, g)–bounding and slaloms (= corsets), see [151], [164, Ch.VI,§2], Goldstern and Shelah =-=[46]-=-, Ros̷lanowski and Shelah [87]. We may consider in the tree version for every limit δ ∈ S (S ⊆ κ is a stationary set) to omit tops for one or just ≤ |δ| branches in a condition p = (T, ¯r), provided f... |

10 |
Uniformization problems and the cofinality of the infinite symmetric groups
- Sharp, Thomas
- 1994
(Show Context)
Citation Context ...ere a bound to γ(G) really better than the first strongly inaccessible > |G| (gotten by Hamkins [54])? Thomas also had started investigating cofinalities of some natural groups, (see Sharp and Thomas =-=[91]-=-, [92], Thomas [177]). He drew me to it and I was particularly glad to see that pcf pops in naturally; e.g., (see Shelah and Thomas [170]) if λn ∈ CF(Sym(ω)) and λ ∈ pcf{λn : n < ω} then λ ∈ CF(Sym(ω)... |

10 |
A theorem and some consistency results in partition calculus
- Shelah, Stanley
- 1987
(Show Context)
Citation Context ...λ = ℵ1, so λ = λ <λ means CH as they choose another extreme case of the unknown). Now, for me a try at consistency of negative answers for 8.6 calls for using historic forcing (see Shelah and Stanley =-=[168]-=-, Ros̷lanowski and Shelah [83, §3]; it is explained below). On the other hand, large cardinals may make some positive results easier. Question 8.8. Assume that λ > κ > |ζ| + σ, and κ is a compact card... |

9 |
and Stevo Todorčević, Partition properties of ω1 compatible with CH, Fund
- Abraham
- 1997
(Show Context)
Citation Context ...ls with NNR solving the specific dichotomy (and really satisfies the [138] promise circumvented in [164, Ch.XVIII, §2]) but left 2.15 open. Eisworth suggested to me (motived by Abraham and Todorčević =-=[4]-=-) Question 2.17. Is the following consistent with ZFC+CH 2 H(λ) is the family of sets with transitive closure of cardinality < λsON WHAT I DO NOT UNDERSTAND 17 (∗) if Aα ∈ [ω1] ℵ0 and α < β ⇒ Aα ⊆ Aβ ... |

9 | Set theoretic real analysis - Ciesielski - 1997 |