THE PROPER FORCING AXIOM, PRIKRY FORCING, AND THE SINGULAR CARDINALS HYPOTHESIS

JUSTIN TATCH MOORE

Abstract. The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at κ then every stationary subset of S_{κ^{+}}^{ω} = {α < κ^{+} : cf(α) = ω} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.

1. Introduction

A stationary subset S of a regular cardinal θ is said to reflect if there is a δ < θ of uncountable cofinality such that S ∩ δ is stationary in δ. Similarly, a family ℱ of stationary sets is said to simultaneously reflect if there is a δ < θ of uncountable cofinality such that S ∩δ is stationary in δ for every S in ℱ. Notice the cofinality of δ acts as an upper bound for the number of disjoint subsets of θ which can simultaneously reflect at δ.

Reflection and simultaneous reflection have been widely studied in set theory with a number of applications to areas such as cardinal arithmetic, descriptive set theory, and infinitary combinatorics. Our starting point will be the following theorem of Foreman, Magidor, and Shelah.

Theorem 1.1. [3] Martin’s Maximum implies that for every uncountable regular cardinal θ > ω_{1} and every collection ℱ of ω_{1} many stationary subsets of S_{θ}^{ω} = {α < θ : cf(α) = ω} there is a δ < θ of cofinality ω_{1} which simultaneously reflects every element of ℱ. Moreover, it can be arranged that the union of ℱ contains a club in δ.

2000 Mathematics Subject Classification. 03E05, 03E10, 03E75.

Key words and phrases. minimal walks, MRP, PFA, Prikry forcing, reflection, SCH.

The research presented in this paper was supported by NSF grant DMS-0401893.

Since for every regular uncountable θ there is a partition of S_{θ}^{ω} into disjoint stationary sets, they conclude that MM implies that θ^{ω_{1}} = θ for all regular θ ≥ ω_{2}. By Silver’s theorem [11] this in turn implies the Singular Cardinals Hypothesis — that 2^{κ} = κ^{+} for every singular strong limit κ.

In this paper, we will introduce and explore a new notion of reflection called trace reflection and prove a result analogous to Theorem 1.1.

Theorem 1.2. (MRP) Suppose that Ω ⊆ S_{θ}^{ω} is a non-reflecting stationary set and that C⃗ avoids Ω. If ℱ is a collection of stationary subsets of Ω and ℱ has size ω_{1} then there is a δ < θ of cofinality ω_{1} such that every element of ℱ simultaneously trace reflects at δ.

It will follow that MRP implies any failure of SCH must occur first at a singular cardinal κ such that every stationary subset of S_{κ^{+}}^{ω} reflects. The above theorem also has the following corollary.

Corollary 1.3. Suppose that M ⊆ V is an inner model with the same cardinals such that for some cardinal κ

(1) cf(κ)^{V} = ω < cf(κ)^{M} = κ and

(2) every stationary subset of κ^{+} in M is stationary in V. Then MRP fails in V. In particular, MRP fails in any generic extension by Prikry forcing.^{1}

This paper is intended to be self contained. Section 2 contains the definition of trace reflection and all of the necessary background on Todorčević’s trace function. Section 3 provides the necessary background on the Set Mapping Reflection Principle which will figure prominently in the analysis. The main results then follow in Section 4.

The notation in the paper is mostly standard. All ordinals are von Neumann ordinals — the set of their predecessors. H(θ) is the collection of all sets of hereditary cardinality less than θ. If X is an uncountable set, [X]^{ω} is used to denote all countable subsets of X. See [5] or [6] for more background; see [4] for some information on the combinatorics of [X]^{ω}, the club filter, and stationary subsets of [X]^{ω}.

I would like to thank the referee for their careful reading useful comments and suggestions.

1When I submitted this paper I was under the impression that it was unknown whether PFA always failed in a Prikry extension. Since the acceptance of this paper I have been made aware that this was not the case. Magidor has shown in an unpublished note that, by a slight modification of an argument of Todorčević, PFA implies □_{κ,ω_{1}} fails for all κ > ω_{1}. On the other hand, Cummings and Schimmerling have shown in [2] that after Prikry forcing at κ, □_{κ,ω} and hence □_{κ,ω_{1}} always holds.

2. Trace Reflection

In this section I will define trace reflection. First recall Todorčević’s notion of a walk on a given cardinal θ (see [12] or [13]). A C-sequence is a sequence C⃗ = ⟨C_{α} : α < θ⟩ where θ is an ordinal, C_{α} is closed and cofinal in α for limit ordinals α, and C_{α+1} = {α}. A C-sequence C⃗ is said to avoid a subset Ω ⊆ θ if C_{α} is disjoint from Ω for every limit α < θ. Notice that if Ω ⊆ θ is a non-reflecting stationary set then there is a C-sequence on θ which avoids Ω. Conversely, any Ω ⊆ θ which is avoided by a C-sequence cannot reflect.

For a given C-sequence, the trace function is defined recursively by

tr(α, α) = {}
tr(α, β) = tr(α, min(C_{β} \ α)) ∪ {β}.

Hence tr(α, β) contains all ordinals “visited” in the walk from β down to α along the C-sequence except for the destination α.^{2} The following property of the trace function captures some of its most important properties.

Fact 2.1. If α < β and α is a limit then there is an α_{0} < α such that

tr(α, β) ⊆ tr(γ, β)

whenever α_{0} < γ < α. If C⃗ avoids {α} then it can further be arranged that

tr(α, β) ∪ {α} ⊆ tr(γ, β).

Proof. First, observe that if ξ is in tr(α, β) then either C_{ξ} ∩ α bounded or else α is in C_{ξ}. Furthermore, the latter can only occur if ξ is the least element of tr(α, β). If α_{0} < α is an upper bound for every set C_{ξ} ∩ α such that ξ is in tr(α, β) and α ∉ C_{ξ}, then it is easily checked that α_{0} has the desired properties (use induction on β). Such a bound exists since tr(α, β) is finite. Finally, if C⃗ avoids {α} then α is not in C_{ξ} for any ξ in tr(α, β). It is therefore possible to prove the stronger conclusion in this case. □

Let θ be an ordinal of uncountable cofinality. For a given C-sequence C⃗ of length θ, define ℋ (C⃗) to be the collection of all X ⊆ θ such that whenever E ⊆ θ is closed and unbounded, there are α < β in E with tr(α, β) ∩ X ≠ ∅. Clearly the complement of ℋ (C⃗) is a σ-ideal.

We say an element X of ℋ (C⃗) trace reflects with respect to C⃗ if there is a δ < θ of uncountable cofinality such that X ∩ δ is in ℋ (C⃗ ↾ δ).

2The omission of the destination is not standard, but it simplifies the presentation at some points. For instance, Fact 2.1 requires this omission.

Simultaneous trace reflection is defined in a similar manner. If C⃗ is clear from the context, I will omit the phrase “with respect to C⃗.” As with ordinary simultaneous reflection, if ℱ is a disjoint family of elements of ℋ (C⃗) which simultaneously trace reflect at δ, then the cardinality of ℱ is at most the cofinality of δ.

3. Set Mapping Reflection

Now I will recall some definitions associated with the Set Mapping Reflection Principle. For the moment let X be a fixed uncountable set and let θ be a regular cardinal such that H(θ) contains [X]^{ω}. The set [X]^{ω} is equipped with a natural topology — the Ellentuck Topology — defined by declaring intervals of the form

[x, N] = {Y ∈ [X]^{ω} : x ⊆ Y ⊆ N}

to be open where x is a finite subset of N. If M is a countable elementary submodel of H(θ) and Σ is a subset of [X]^{ω} then we say that Σ is M-stationary if E ∩ Σ ∩ M is non-empty whenever E ⊆ [X]^{ω} is a closed unbounded set in M. If Σ is set mapping defined on a collection of countable elementary submodels of H(θ) then we say that Σ is open stationary if Σ(M) ⊆ [X]^{ω} is open and M-stationary for all relevant M.

A set mapping Σ as above reflects if there is a continuous ∈-chain ⟨N_{ν} : ν < ω_{1}⟩ in the domain of Σ such that for every limit ν > 0, N_{ξ} ∩ X is in Σ(N_{ν}) for coboundedly many ξ in ν. If this happens then ⟨N_{ν} : ν < ω_{1}⟩ is called a reflecting sequence for Σ. The axiom MRP asserts that every open stationary set mapping defined on a club reflects. In [8] it is shown that MRP is a consequence of PFA. It is also shown there that it implies 2^{ℵ_{0}} = 2^{ℵ_{1}} = ℵ_{2} and that □(κ) fails for all regular κ > ω_{1}.

4. The main results

We now proceed to the proof of the main theorem.

Theorem. (MRP) Suppose that Ω ⊆ S_{θ}^{ω} is a non-reflecting stationary set and that C⃗ avoids Ω. If ℱ is a collection of stationary subsets of Ω and ℱ has size ω_{1} then there is a δ < θ of cofinality ω_{1} such that every element of ℱ simultaneously trace reflects at δ.

Proof. Let ℱ = {Ω_{ξ} : ξ < ω_{1}} be given and let {S_{ξ} : ξ < ω_{1}} be a sequence of disjoint stationary sets such that ξ < min(S_{ξ}) and ⋃_{ξ<ω_{1}} S_{ξ} contains a club. For M a countable elementary submodel of H(2^{θ}^{+}) which contains ℱ, define Σ_{ℱ} (M) to be the collection of all countable N ⊆ θ such that either N ∩ θ has a last element or else sup N < sup(M ∩ θ) and

tr(sup N, sup(M ∩ θ)) ∩ Ω_{δ} ≠ ∅

where δ is such that M ∩ ω_{1} is in S_{δ}. That Σ_{ℱ} (M) is open is a consequence of Fact 2.1.

Claim 4.1. Σ_{ℱ} (M) is M-stationary.

Proof. Let E ∈ M be a club of countable subsets of θ and let δ be such that M ∩ ω_{1} is in S_{δ}. By elementarity and assumption that Ω_{δ} is stationary, there is an α in Ω_{δ} ∩ M such that for every α_{0} < α, there is an N in E ∩ M such that α_{0} < sup(N) < α. By Fact 2.1 it is possible to find an N in E ∩ M such that α is in tr(sup(N), sup(M ∩ θ)). □

Now, let ⟨N_{ξ} : ξ < ω_{1}⟩ be a reflecting sequence for Σ_{ℱ} and put

E = {sup(N_{ξ} ∩ θ) : ξ < ω_{1}}
δ = sup E.

It suffices to show that for every ξ < ω_{1} and closed unbounded E′ ⊆ δ that there are α < β in E′ such that tr(α, β) ∩ Ω_{ξ} is non-empty. Let β < δ be a limit point of E ∩E′ such that N_{ν} ∩ω_{1} is in S_{ξ} where ν < ω_{1} is such that β = sup(N_{ν} ∩ θ). By virtue of ⟨N_{ξ} : ξ < ω_{1}⟩ reflecting Σ_{ℱ} and the definition of E, there is a β_{0} < β such that if α is in E with β_{0} < α < β then tr(α, β) ∩ Ω_{ξ} is non-empty. Selecting α in E ∩ E′ with β_{0} < α < β, we now have α < β both in E′ with tr(α, β) ∩ Ω_{ξ} non-empty as desired. □

We finish the section with proof of the corollary.

Corollary. Suppose that M ⊆ V is an inner model with the same cardinals such that for some cardinal κ

(1) cf(κ)^{V} = ω < cf(κ)^{M} = κ and

(2) every stationary subset of κ^{+} in M is stationary in V. Then MRP fails in V. In particular, MRP fails in any generic extension by Prikry forcing.

Proof. Let C⃗ be a C-sequence in M of length θ = κ^{+} such that for every α < θ, C_{α} has ordertype at most κ and C⃗ avoids

Ω = {α < θ : cf(α)^{M} = κ}.

Let {Ω_{ξ} : ξ < κ} be a partition in M of Ω into disjoint stationary sets. Pick an X ⊆ κ in V which is countable and cofinal in κ. Now suppose towards a contradiction that MRP holds in V. By the main theorem there would be a δ < θ of cofinality ω_{1} such that Ω_{ξ} trace reflects at δ for every ξ in X. Now observe that the cofinality of δ must be less than κ in M since otherwise it would have countable cofinality in V. Let E ⊆ δ be closed and unbounded with |E| < κ and E in M. Put

X^{∗} = {ξ < κ : ∃α, β ∈ E(tr(α, β) ∩ Ω_{ξ} ≠ ∅)}.

Certainly X^{∗} is in M, has size less than κ (since the {Ω_{ξ} : ξ < κ} are all pairwise disjoint), and is cofinal (since it contains X). But this is a contradiction since κ is regular in M. □

5. Concluding remarks

Assaf Sharon has announced that SCH can fail at κ (even for κ = ℵ_{ω}) and yet every stationary subset of κ^{+} reflects. Hence it is not possible to prove the conjecture by establishing a ZFC connection between the existence of a non-reflecting stationary subset of κ^{+} and the failure of SCH at κ. Since the acceptance of this paper, Viale has shown that MRP also fails in extensions by forcings which have the weak Prikry property [14]. This class of forcings includes the forcing used by Sharon.

A different approach is to try to replace the assumption of a nonreflecting stationary subset of S_{κ^{+}}^{ω} with the existence of a good scale for κ. The motivating factor is that a good scale for κ always exists if SCH fails first at κ (see Main Claim 1.3, p 46 in [9]). One could attempt to refute the existence of good scales using MRP and thus prove MRP implies SCH (Magidor has shown that MM implies good scales do not exist). This is not possible,^{3} however, since Magidor has shown from appropriate large cardinal assumptions that PFA is consistent with □^{∗}(κ) holding at all cardinals κ above ω_{2} [7]. Since □^{∗}(κ) implies that there is a good scale for κ [1], this approach will not work.

Finally, there are some results which link reflection in [λ]^{ω} to SCH. In [15] Veličković showed that if θ > ω_{1} is regular and stationary subsets of [θ]^{ω} reflect to an internally closed unbounded set (strongly reflect in the language of [15]) then θ^{ω} = θ. An immediate consequence is that PFA^{+} implies SCH. Recently Shelah improved this result by showing that reflection of stationary subsets of [θ]^{ω} to sets of size ω_{1} is already sufficient to deduce θ^{ω} = θ [10].

References

[1] James Cummings, Matthew Foreman, and Menachem Magidor. Squares, scales and stationary reflection. J. Math. Logic, 1(1):35–98, 2001.

[2] James Cummings and Ernest Schimmerling. Indexed squares. Israel J. Math., 131:61–99, 2002.

3I would like to thank John Krueger for pointing this out shortly before publication.

[3] Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s Maximum, saturated ideals, and nonregular ultrafilters. I. Ann. of Math. (2), 127(1):1–47, 1988.

[4] T. Jech. Multiple forcing, volume 88 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1986.

[5] Thomas Jech. Set theory. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, second edition, 1997.

[6] Kenneth Kunen. An introduction to independence proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1983.

[7] Menachem Magidor. Lectures on weak square principles and forcing axioms. Jerusalem logic seminar, 1995.

[8] Justin Tatch Moore. Set mapping reflection. J. Math. Logic, 5(1):87–98, 2005.

[9] Saharon Shelah. Cardinal arithmetic, volume 29 of Oxford Logic Guides. The Clarendon Press Oxford University Press, New York, 1994. Oxford Science Publications.

[10] Saharon Shelah. Reflection implies SCH. Preprint, July 2004.

[11] Jack Silver. On the singular cardinals problem. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pages 265–268. Canad. Math. Congress, Montreal, Que., 1975.

[12] Stevo Todorčević. Partitioning pairs of countable ordinals. Acta Math., 159(3– 4):261–294, 1987.

[13] Stevo Todorčević. Coherent sequences. In Handbook of Set Theory. North-Holland, (in preparation).

[14] Matteo Viale. Prikry type forcings, the Singular Cardinals Hypothesis and the Proper Forcing Axiom. preprint, Aug. 2005.

[15] Boban Veličković. Forcing axioms and stationary sets. Adv. Math., 94(2):256– 284, 1992.