Collapsing functions

Abstract

We define what it means for a function on ω1 to be a collapsing function for λ and show that if there exists a collapsing function for (2 ω 1 ) + , then there is no precipitous ideal on ω1. We show that a collapsing function for ω2 can be added by forcing. We define what it means to be a weakly ω1-Erdös cardinal and show that in L[E], there is a collapsing function for λ iff λ is less than the least weakly ω1-Erdös cardinal. As a corollary to our results and a theorem of Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of precipitous ideals on ω1. We also show that the following statements hold in L [E]. The least cardinal λ with the Chang property (λ, ω1) (ω1, ω) is equal to the least ω1-Erdös cardinal. In particular, if j is a generic elementary embedding that arises from non-stationary tower forcing up to a Woodin cardinal, then the minimum possible value of j(ω1) is the least ω1-Erdös cardinal.One of the striking consequences of large cardinals is that they imply the existence of a generic elementary embedding j : V −→ M with M transitive and crit(j) = ω 1 . For example, if δ is a Woodin cardinal, then there is a condition in the non-stationary tower P <δ that forces the existence of such an embedding (see [6]). The value of j(ω 1 ) tends to be rather large. For example, if δ is a Woodin cardinal in an iterable extender model L[E], then forcing with P L[E] <δ over L[E] produces an embedding j with j(ω 1 ) ≥ the least ω 1 -Erdös cardinal of L[E].It is natural to ask if large cardinals imply the existence of a precipitous ideal on ω 1 since this would imply the existence of a generic elementary embedding j with j(ω 1 ) < (2 ω1 ) + . One way to disprove this might be to show that there is a set forcing which kills all precipitous ideals on ω 1 . In this paper we present some partial results on this question and, in particular, show that Woodin limits of Woodin cardinals do not imply the existence of precipitous ideals on ω 1 .The contents of the paper are as follows. First we define what it means for there to exist a collapsing function for a cardinal λ. Then we show that if there exists a collapsing function for (2 ω1 ) + , then no ideal on ω 1 is precipitous. It is easy, as we show, to add a collapsing function for ω 2 by forcing; whether this can be done for ω 3 is not known.Next we turn to inner models of the form L [E], where E is a coherent sequence of extenders. We define what it means for a cardinal to be weakly ω 1 -Erdös. Then we show that in L[E], there is a collapsing function on λ if and only if λ is less than the least weakly ω 1 -Erdös cardinal. In particular, in L [E], there are no precipitous ideas on ω 1 . As a corollary to this and an earlier result of Itay Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of a precipitous ideal on ω 1 .We conjecture that in L[E], there is a precipitous ideal on κ if and only if κ is measurable. John Steel [5] has shown this by a method different from ours under hypotheses more rest...

SET FORCING AND STRONG CONDENSATION FORH(ω2)

“…For example, the theory of Lκ [R]. 11 It will be clear that P 2 is a totally proper forcing notion, i.e., a proper forcing that does not add reals. However, the iterants of P 2 are not proper, and thus we will directly deal with P 2 rather than adopting the general framework of proper forcing.…”

A quasi-lower bound on the consistency strength of PFA

“…It is known, using collapsing functions, that Strong Condensation for ω 3 implies that there is no precipitous ideal on ω 1 (see [15]), which makes Strong Condensation for ω 3 already a most interesting property. For every cardinal α, Strong Condensation implies Strong Condensation up to α implies Strong Condensation for α.…”

“…Then ξ can be defined using finite sets of parameters S 0 ⊆ sup(S-supp(r) ∩ θ) and 15 It follows that ξ < sup(S-supp(j(t)) ∩ θ) < sup(S-supp(r) ∩ θ), which is equal to sup r * * θ by the usual arguments. For the first statement, assume ξ ∈ M θ , ξ < θ.…”

“…15 The Skolem Hull in H j −1 (ν) is taken with respect to the j-pre-image of the wellorder of H ν M chosen above. By induction on θ.…”

On almost precipitous ideals

“…Woodin following Foreman, Magidor and Shelah [3] showed that Col(ℵ 1 , δ) turns N S ℵ 1 into a presaturated ideal. On the other hand Schimmerling and Velickovic [8] showed that there is no precipitous ideals on ℵ 1 in L [E] up to at least a Woodin limit of Woodins. Also by [8], there is f :…”