Saturated ideals

Kunen, Kenneth

doi:10.2307/2271949

Cited by 136 publications

(76 citation statements)

References 12 publications

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“…We force from large cardinals a model of Δ ℵ 2 ,ℵ 2 +1 where the tree property at ℵ 2 +1 fails, thus we demonstrate Theorem 1.3. The proof combines Magidor and Shelah's technique for getting Δ ℵ 2 ,ℵ 2 +1 with some ideas from Kunen's paper (see [2]). …”

Section: Note That the Setmentioning

confidence: 99%

Reflection of Stationary Sets and the Tree Property at the Successor of a Singular Cardinal

Fontanella¹,

Magidor²

2017

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Abstract. We show that from infinitely many supercompact cardinals one can force a model of ZFC where both the tree property and the stationary reflection hold at ℵ 2 +1 . §1. Introduction. One of the most fruitful research areas in set theory concerns the investigation of models of set theory where properties of large cardinals hold at small cardinals. The results presented in this paper focus on two properties of weakly compact cardinals that, under large cardinal assumptions, can be forced at small cardinals, the tree property and the reflection of stationary sets. To get a model of the tree property at the double successor of a regular cardinal it is enough to assume the consistency of a weakly compact cardinal and force with a classical poset due to Mitchell (see [6]). Forcing the tree property at the successor of a singular cardinal is harder and requires much stronger assumptions. The first model of the tree property at the successor of a singular cardinal, was defined by Magidor and Shelah in [4] who proved from large cardinals the consistency of the tree property at ℵ +1 . The hypotheses used in such result have the consistency strength of a large cardinal between a huge cardinal and a 2-huge cardinal; this was later improved by Sinapova [10] who was able to force the tree property at ℵ +1 from weaker large cardinal assumptions, namely assuming the consistency of infinitely many supercompact cardinals. Another paper by Neeman [7] shows that, assuming the consistency of infinitely many supercompact cardinals, one can force a model where the tree property holds simultaneously at ℵ +1 and at every ℵ n with n ≥ 2. All these constructions can be adapted to force the tree property at the successor of any singular cardinal κ of countable cofinality, however in all these models the reflection of stationary sets fails at κ + . In the case of ℵ +1 , for instance, these forcing constructions all add a bad scale. So it is natural to ask whether the tree property and the reflection of stationary sets are incompatible at the successor of a singular cardinal. In this paper we answer this question for a particular cardinal, namely ℵ 2 +1 . We show that, assuming the consistency of infinitely many supercompact cardinals, one can force a model where both the tree property and the stationary

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Section: Note That the Setmentioning

confidence: 99%

Reflection of Stationary Sets and the Tree Property at the Successor of a Singular Cardinal

Fontanella¹,

Magidor²

2017

J. symb. log.

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“…The theory of saturation of ideals has a long record of intimacy with that of large cardinals, for the details of which one should consult Kunen [5]. The best relative consistency results between the theories of saturated ideals on u>x and large cardinals are the theorem of Kunen [5] that the consistency of ZFC + "there is a huge cardinal" implies that of ZFC + "there is an co2-saturated ideal on co,", and the result of Mitchell [7] that the consistency of ZFC + "there is an co2-saturated ideal on co," implies that of ZFC + "there is a cardinal k which is 52(«)-measurable".…”

Section: Proof Suppose P Lh Vx G R 3 Vmentioning

confidence: 99%

Two Consequences of Determinacy Consistent with Choice

Steel¹,

Wesep²

1982

Transactions of the American Mathematical Society

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Abstract. We begin with a ground model satisfying ZF + AD + ACR, and from it construct a generic extension satisfying ZFC + ô2 = w2 + "me nonstationary ideal on w | is w2-saturated". 0. Introduction. The "axiom" of determinacy (henceforth AD) implies many interesting propositions about small cardinal numbers, for example 2, and the second that the nonstationary ideal on w, is w2-saturated. We shall show that the theory consisting of ZFC + ADL(R) together with these two propositions is consistent. (ADL(R) is the assertion that all games in £(R), a large class of definable games, are determined.)In consonance with our working hypothesis, we would like to prove our result assuming only the consistency of ZFC + ADL(R), or equivalently, that of ZF + AD + DC (cf. §1). At present we cannot do this. We instead assume the consistency of ZF + AD + ACR, where ACR is the axiom of choice for families indexed by the reals. Consistency-wise, ZF + AD + ACR is one of the strongest theories known to man; we show in §1 that it proves the consistency of ZF + AD + DC. The only upper bound we know on the consistency strength of ZF + AD + ACR is that of ZF + ADR + "Ö is regular". We offer some partial justification for our use of such a strong hypothesis at the end of the paper.Our consistency proof is by forcing. We start with a ground model M satisfying ZF + AC + ACR. By a two-step iteration we construct a generic extension M [C7] "onto M so that z\h (h: w2 -* R ). The forcing will add no new reals and preserve w2, i.e.,We then pass t0 the inner model N _ £(RM ^ an(J show it has the desired properties. The paper is organized as follows: In §1 we give some preliminaries and background information. In particular, we set forth those consequences of AD whose

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“…The argument as presented here is self-contained, but is motivated by the Kunen construction in [2].…”

Section: Hence In M[g)i"2f Is Of Typementioning

confidence: 99%

“…For each partial order P, we let cffi(P) be the complete Boolean algebra corresponding to P. (2) for all x E P, 7T(e(x» = x, (3) for ally E Q, e(7T(y»;;;, y, (4) Suppose e: Pc-> Q. Let S be a separative partial ordering in V p .…”

Section: Hence In M[g)i"2f Is Of Typementioning

confidence: 99%