Chapter 6 - Magidor Forcing

In 1978 Magidor [37] constructed the first forcing that changes the cofinality of a cardinal $\kappa$ to an uncountable value $\delta <\kappa$ without collapsing cardinals. His initial assumption was a sequence $⟨U_{\gamma }\mid \gamma <\delta ⟩$ of normal measures on $\kappa$, increasing in the Mitchell order — in modern terms, the top row of a coherent sequence with $o(\kappa )=\delta$. We present the forcing twice: first as a cone of the forcing ${\mathbb{P}}_{\vec{U}}$ of Chapter 5, where all of its properties are inherited for free; then in Magidor’s original direct form, which is the version used in applications. We then explain why the Mitchell-order hypothesis suffices here, in contrast with the Radin forcing ${\mathbb{R}}_{\vec{V}}$, and discuss the optimality of that hypothesis.

6.1 Magidor forcing as a cone of ${\mathbb{P}}_{\vec{U}}$

Fix a coherent sequence of measures $\vec{U}$ with ${\ell }^{\vec{U}}=\kappa +1$ and $o^{\vec{U}}(\kappa )=\delta$, where $0<\delta <\kappa$ (Definition 5.1.1). Recall the splitting of Lemma 5.3.1: the sets $Y_{\alpha }=\{\nu <\kappa \mid o^{\vec{U}}(\nu )=\alpha \},\alpha <\delta ,$ are pairwise disjoint and $Y_{\alpha }\in U(\kappa ,\alpha )$. In particular ${\bigcup }_{\alpha <\delta }{Y}_{\alpha }\in \bigcap \vec{U}(\kappa )$.

Definition 6.1.1 (Gitik, p. 1419).

The Magidor forcing associated with $\vec{U}$ is the cone ${\mathbb{M}}_{\vec{U}}={\mathbb{P}}_{\vec{U}}\upharpoonright ⟨⟨\kappa ,{\bigcup }_{\alpha <\delta }{Y}_{\alpha }⟩⟩,$ i.e. the forcing ${\mathbb{P}}_{\vec{U}}$ (Definitions 5.2.1–5.2.3) below the displayed condition, with the induced orders $\le$ and ${\le }^{\ast }$.

Since the $Y_{\alpha }$‘s are disjoint, every point $\nu$ appearing in a stem of a condition in ${\mathbb{M}}_{\vec{U}}$ carries a well-defined index, namely $\alpha =o^{\vec{U}}(\nu )$: ordinal entries $d_m$ (case (3a) of Definition 5.2.1) are the points of index $0$, and a pair $⟨\nu ,A_{\nu }⟩$ is a point of index $o^{\vec{U}}(\nu )>0$. A stem is therefore nothing but a finite increasing sequence of indexed points $⟨({\nu }_1,{\alpha }_1),\dots ,({\nu }_n,{\alpha }_n)⟩$ together with measure-one sets — which is exactly the shape of Magidor’s original conditions.

Lemma 6.1.2 (Canonical refinement).

Every condition of ${\mathbb{M}}_{\vec{U}}$ has a direct extension $p$ such that

1. the top measure-one set $B$ of $p$ satisfies $B\subseteq {\bigcup }_{\alpha <\delta }{Y}_{\alpha }$, and
2. for every pair $⟨\nu ,A_{\nu }⟩$ in the stem of $p$, $A_{\nu }\subseteq {\bigcup }_{\beta <o^{\vec{U}}(\nu )}{Y}_{\beta }\cap \nu$.

Proof. Both shrinkings are legitimate for ${\le }^{\ast }$: for (2), apply Lemma 5.3.1 at $\nu$ instead of $\kappa$ — the set $Y_{\beta }\cap \nu =\{\mu <\nu \mid o^{\vec{U}}(\mu )=\beta \}$ lies in $U(\nu ,\beta )$ for each $\beta <o^{\vec{U}}(\nu )$, and $A_{\nu }\in {\bigcap }_{\beta <o^{\vec{U}}(\nu )}U(\nu ,\beta )$; intersecting finitely many measure-one sets stays in the intersection. For (1) there is nothing to do beyond the cone condition itself. $■$

The point of the refinement is this: below $p$, any point inserted into the block of a pair $⟨\nu ,A_{\nu }⟩$ lies in $A_{\nu }$, hence has index $<o^{\vec{U}}(\nu )$. So the block below a stem point of index $\alpha$ only ever acquires points of index $<\alpha$ — a self-similar, downward-nested structure governed entirely by the indices. Magidor’s original definition builds this in from the start by recording the index of each stem point explicitly.

6.2 Magidor’s original definition

Let $\delta <\kappa$ and let $\vec{U}=⟨U_{\gamma }\mid \gamma <\delta ⟩$ be a sequence of normal measures on $\kappa$, increasing in the Mitchell order $⊲$. (Such a sequence exists iff $o(\kappa )\ge \delta$; it is the top row $\vec{U}(\kappa )$ of any coherent sequence with $o^{\vec{U}}(\kappa )=\delta$.) For $\mu <\nu <\delta$, choose a function $f_{\mu }^{\nu }:\kappa \to V$ representing $U_{\mu }$ in the ultrapower by $U_{\nu }$: $[f_{\mu }^{\nu }{]}_{U_{\nu }}=U_{\mu }.$ Such functions exist precisely because $U_{\mu }⊲U_{\nu }$ means $U_{\mu }\in \mathrm{Ult}(V,U_{\nu })$. We write $\vec{f}=⟨f_{\mu }^{\nu }\mid \mu <\nu <\delta ⟩$.

The functions $f_{\mu }^{\nu }$ allow the measures below a point $\rho <\kappa$ to be decoded: for $U_{\nu }$-almost all $\rho$, the objects $f_{\mu }^{\nu }(\rho )$ are normal measures on $\rho$ which cohere with the $f$‘s exactly as the $U_{\mu }$‘s cohere among themselves. This is the content of the next lemma, due to Magidor.

; the good sets).

For $0<\gamma <\delta$, the following sets belong to $U_{\gamma }$: $A_{\gamma }=\{\rho <\kappa \mid \forall \mu <\nu <\gamma f_{\mu }^{\gamma }(\rho )⊲f_{\nu }^{\gamma }(\rho )\text{are normal ultrafilters over }\rho \},$ $B_{\gamma }=\{\rho \in A_{\gamma }\mid \forall \mu <\nu <\gamma [f_{\mu }^{\nu }\upharpoonright \rho {]}_{f_{\nu }^{\gamma }(\rho )}=f_{\mu }^{\gamma }(\rho )\},$ and $B_0=\{\rho <\kappa \mid \rho \text{ is inaccessible}\}\in U_0$.

Proof sketch. Both statements are Łoś computations in $M_{\gamma }=\mathrm{Ult}(V,U_{\gamma })$. For $\rho =\kappa$, the defining clauses of $A_{\gamma }$ read ”$f_{\mu }^{\gamma }(\kappa {)}^{M_{\gamma }}⊲f_{\nu }^{\gamma }(\kappa {)}^{M_{\gamma }}$ are normal ultrafilters over $\kappa$”, i.e. ”$U_{\mu }⊲U_{\nu }$ are normal ultrafilters over $\kappa$”, which hold in $M_{\gamma }$ since the $U_{\mu }$‘s for $\mu <\gamma$ are elements of $M_{\gamma }$ by the choice of the representing functions. Similarly, the clause defining $B_{\gamma }$ at $\rho =\kappa$ reads $[j_{\gamma }(f_{\mu }^{\nu })\upharpoonright \kappa {]}_{U_{\nu }}=[f_{\mu }^{\nu }{]}_{U_{\nu }}=U_{\mu }=[f_{\mu }^{\gamma }{]}_{U_{\gamma }}$, again true in $M_{\gamma }$. We leave the verification of the details as an exercise. $■$

For $\rho \in B_{\gamma }$, the measures $f_{\mu }^{\gamma }(\rho )$ ($\mu <\gamma$) on $\rho$ are the exact analogue of the sequence $\vec{U}(\rho )$ of Chapter 5 — obtained here by explicit decoding rather than by coherence. We can now define the forcing, following Magidor [37]; our notation follows the restatement by Fuchs.

Definition 6.2.2 (Neighbor functions).

For a finite $a\subseteq \delta$ define $l_a:\delta \to \delta \cup \{-1\}$ and $r_a:\delta \to \delta +1$ by $l_a(\gamma )=\max (a\cap \gamma )(\text{or }-1\text{ if }a\cap \gamma =\varnothing ),r_a(\gamma )=\min ((a\cup \{\delta \})\setminus (\gamma +1)),$ the nearest neighbors of $\gamma$ inside $a$, to the left and to the right.

).

The Magidor forcing $\mathbb{M}(\vec{U},\vec{f})$ consists of pairs $⟨g,G⟩$ such that

1. $\mathrm{dom}(g)$ is a finite subset of $\delta$, and $\mathrm{dom}(G)=\delta \setminus \mathrm{dom}(g)$;
2. for all $\gamma \in \mathrm{dom}(g)$, $g(\gamma )\in B_{\gamma }$, and $g$ is strictly increasing;
3. for all $\gamma \in \mathrm{dom}(G)$, if $\theta =r_{\mathrm{dom}(g)}(\gamma )<\delta$ then $G(\gamma )\in f_{\gamma }^{\theta }(g(\theta ))$, and if $\theta =\delta$ then $G(\gamma )\in U_{\gamma }$;
4. if $\gamma <\xi <\delta$ with $\gamma \in \mathrm{dom}(g)$ and $\xi \in \mathrm{dom}(G)$, then $g(\gamma )\cap G(\xi )=\varnothing$.

The ordering is defined by $⟨g^{\prime },G^{\prime }⟩\le ⟨g,G⟩$ ($g^{\prime }$ stronger) iff

1. $g\subseteq g^{\prime }$;
2. $G^{\prime }(\gamma )\subseteq G(\gamma )$ for all $\gamma \in \mathrm{dom}(G^{\prime })$;
3. $g^{\prime }(\gamma )\in G(\gamma )$ for all $\gamma \in \mathrm{dom}(g^{\prime })\setminus \mathrm{dom}(g)$.

We write $⟨g^{\prime },G^{\prime }⟩{\le }^{\ast }⟨g,G⟩$ (direct extension) iff $⟨g^{\prime },G^{\prime }⟩\le ⟨g,G⟩$ and $g^{\prime }=g$.

Thus $g$ is the stem: a finite increasing sequence of points, each tagged with its index $\gamma$, chosen from the good set $B_{\gamma }$. Clause (3) says that a measure-one set at an unused index $\gamma$ is measured by $f_{\gamma }^{\theta }(g(\theta ))$ — the $\gamma$-th measure on $\rho =g(\theta )$, where $(\rho ,\theta )$ is the nearest stem point above $\gamma$; if there is none, the ambient measure $U_{\gamma }$ is used. Clause (4) guarantees that a new point inserted at index $\gamma$ lands in the interval $(g(l(\gamma )),g(r(\gamma )))$ determined by its stem neighbors, so that $g^{\prime }$ remains strictly increasing. Note that, in parallel with the rest of these notes, $\le$ means “stronger than”; Magidor’s and Gitik’s papers use the opposite convention. In some presentations one additionally fixes an ordinal $\tilde{\alpha }<\kappa$ and requires all stem points to exceed $\tilde{\alpha }$ (the “forcing above $\tilde{\alpha }$”); this freedom is useful in iteration arguments but plays no role here.

Definition 6.2.4.

Let $G\subseteq \mathbb{M}(\vec{U},\vec{f})$ be generic over $V$. The Magidor sequence is $c_G(\gamma )=g(\gamma )\text{for the unique value forced by any }⟨g,G⟩\in G\text{ with }\gamma \in \mathrm{dom}(g).$

Lemma 6.2.5.

The Magidor sequence $c_G:\delta \to \kappa$ is defined on all of $\delta$, strictly increasing, continuous at limit ordinals, and cofinal in $\kappa$. In particular its range is a closed unbounded subset of $\kappa$ of order type $\delta$ (for $\delta$ a limit ordinal).

Proof sketch. Totality is a density argument: given $⟨g,G⟩$ and $\gamma \notin \mathrm{dom}(g)$, with $\theta =r_{\mathrm{dom}(g)}(\gamma )$, any $\rho \in G(\gamma )$ above the current stem values below $\gamma$ yields a stronger condition $⟨g\cup \{⟨\gamma ,\rho ⟩\},G^{\prime }⟩$ — the side conditions of Definition 6.2.3 hold by clauses (3) and (4) for $⟨g,G⟩$. Monotonicity is built into clause (2). Cofinality: for $\gamma$ with no stem point above, $G(\gamma )\in U_{\gamma }$ is unbounded in $\kappa$, so values at index $\gamma$ are forced unboundedly high. Continuity at a limit $\lambda$: if $\lambda \in \mathrm{dom}(g)$, then for every $\xi \in (l_{\mathrm{dom}(g)}(\lambda ),\lambda )$ the set $G(\xi )$ is measure one for the normal ultrafilter $f_{\xi }^{\lambda }(g(\lambda ))$ over $g(\lambda )$, hence unbounded in $g(\lambda )$; inserting points at larger and larger indices below $\lambda$ therefore forces ${\sup }_{\gamma <\lambda }{c}_G(\gamma )=c_G(\lambda )$. A continuous strictly increasing function has closed range, and a strictly increasing function on $\delta$ has range of order type $\delta$. $■$

Exercise 6.2.6.

Spell out the correspondence between $\mathbb{M}(\vec{U},\vec{f})$ and the cone ${\mathbb{M}}_{\vec{U}}$ of Definition 6.1.1: given a condition $⟨g,G⟩$ with $\mathrm{dom}(g)=\{{\gamma }_1<\cdots <{\gamma }_n\}$, describe the associated condition of ${\mathbb{P}}_{\vec{U}}$ whose stem pairs are $⟨g({\gamma }_i),A_{g({\gamma }_i)}⟩$ with $A_{g({\gamma }_i)}={\bigcap }_{\xi <{\gamma }_i}{G}^{\prime }(\xi )$, where the $G^{\prime }(\xi )$ are the appropriate shrinkings of the $G(\xi )$‘s. Show that the values of the Magidor sequence are exactly the pair coordinates appearing in stems of the generic for the cone, and conclude (using Lemma 3.1.3 and Lemma 6.2.5) that $\mathrm{ran}(c_G)$ is club in $\kappa$ and computes $\mathrm{cf}(\kappa )$ in the extension.

6.3 The main theorem

).

Let $\delta <\kappa$ and let $\vec{U}=⟨U_{\gamma }\mid \gamma <\delta ⟩$ be a $⊲$-increasing sequence of normal measures on $\kappa$. Set $\mathbb{M}=\mathbb{M}(\vec{U},\vec{f})$. Then:

1. $⟨\mathbb{M},\le ,{\le }^{\ast }⟩$ has the Prikry property, and ${\le }^{\ast }$ is $\kappa$-closed;
2. $\mathbb{M}$ has the ${\kappa }^{+}$-chain condition;
3. $\mathbb{M}$ preserves all cardinals and adds no bounded subsets of $\kappa$;
4. ${⊩}_{\mathbb{M}}\mathrm{cf}(\kappa )=\mathrm{cf}(\delta {)}^V$.

Proof. All parts are inherited from the coherent-sequence forcing of Chapter 5 via the cone presentation of Section 6.1. Extend $\vec{U}$ to a coherent sequence with $o^{\vec{U}}(\kappa )=\delta$ (possible since $o(\kappa )\ge \delta$; see the discussion after Definition 5.1.1). The cone ${\mathbb{M}}_{\vec{U}}$ is a subordering of ${\mathbb{P}}_{\vec{U}}$, so the transfer theorem Theorem 5.3.2 applies to it directly:

1. The Prikry property is the ${\mathbb{P}}_{\vec{U}}$-version of Lemma 2.4.4, and the $\kappa$-closure of ${\le }^{\ast }$ is that of Lemma 2.4.3 — in the cone the argument only simplifies, since measure-one sets consist of ordinals.
2. The ${\kappa }^{+}$-c.c. is Lemma 2.4.1: conditions are finite sequences of objects from $V_{\kappa +1}$ of size $\le \kappa$ after fixing the stem, so any two conditions with the same stem are compatible.
3. Cardinal preservation now follows by the standard argument of Theorem 2.4.5: the ${\kappa }^{+}$-c.c. preserves cardinals $\ge {\kappa }^{+}$, and the Prikry property together with $\kappa$-closure of ${\le }^{\ast }$ shows no new bounded subsets of $\kappa$ are added, preserving cardinals $\le \kappa$.
4. By Lemma 6.2.5, $c_G:\delta \to \kappa$ is continuous and cofinal, so ${\mathrm{cf}}^{V[G]}(\kappa )={\mathrm{cf}}^{V[G]}(\delta )$. Since $\delta <\kappa$ and no bounded subsets of $\kappa$ are added, ${\mathrm{cf}}^{V[G]}(\delta )={\mathrm{cf}}^V(\delta )$. (Equivalently, in the cone presentation, the order-type analysis of Theorem 3.2.5 gives $\mathrm{cf}(\kappa )=\mathrm{cf}({\omega }^{\delta })=\mathrm{cf}(\delta )$ directly.) $■$

Exercise 6.3.2 (guided).

Prove parts (1) and (2) of Theorem 6.3.1 directly from Definition 6.2.3, without passing through ${\mathbb{P}}_{\vec{U}}$. Hints: for the closure of ${\le }^{\ast }$, intersect fewer than $\kappa$ many measure-one sets index by index (each $f_{\gamma }^{\theta }(g(\theta ))$ is $g(\theta )$-complete and there are only $|\delta |<\kappa$ many indices); for the Prikry property, imitate the proof of Lemma 2.4.4 with the stem $g$ in place of $d_1,\dots ,d_n$ — the relevant partition argument takes place, for each stem point $g(\theta )$, inside the measures $f_{\gamma }^{\theta }(g(\theta ))$ for $\gamma <\theta$.

6.4 Why Mitchell order suffices: comparison with ${\mathbb{R}}_{\vec{V}}$

It is worth pausing on exactly why Magidor forcing runs on the cheap hypothesis $o(\kappa )=\delta$, while the parallel cofinality change via ${\mathbb{R}}_{\vec{V}}$ requires a measure sequence of length $\delta$ in $\mathcal{A}$ — already length $2$ costs a μ-measurable cardinal (Definition 1.6.3), and length $\ge {\kappa }^{+}$ costs much more (Lemma 1.5.1).

1. The measures concentrate on ordinals. Every measure used in $\mathbb{M}(\vec{U},\vec{f})$ — the $U_{\gamma }$‘s, the decoded measures $f_{\gamma }^{\theta }(\rho )$ — is an ultrafilter over an ordinal, and measure-one sets consist of ordinals. In ${\mathbb{R}}_{\vec{V}}$, by contrast, a measure-one set consists of pairs $⟨\nu ,{\vec{F}}_{\nu }⟩$, i.e. of elements of $V_{\kappa }$ carrying their own measure sequences.
2. Stem points do not spawn nested blocks of new measure sequences. Inserting a point $\rho$ at index $\gamma$ into a Magidor stem commits us to nothing below $\rho$ except the measures $f_{\mu }^{\gamma }(\rho )$ for $\mu <\gamma$, all of which are decoded from the ambient functions $f$ — the recursion bottoms out at ordinals. In ${\mathbb{R}}_{\vec{V}}$, inserting $⟨\nu ,{\vec{F}}_{\nu }⟩$ imports the whole sequence ${\vec{F}}_{\nu }$, whose own measure-one sets again consist of pairs, and so on; this is why the membership $⟨\nu ,{\vec{F}}_{\nu }⟩\in \mathcal{A}$ had to be certified by a constructing embedding (Definition 1.4.2).
3. The strength accounting. A $⊲$-increasing sequence of length $\delta$ over $\kappa$ exists as soon as $o(\kappa )\ge \delta$ — pure Mitchell order, no extenders; by Mitchell’s core model theory [43], the assertion "$o(\kappa )=\delta$" is equiconsistent with its forcing consequences. The measure sequences of Chapter 1 live strictly higher: μ-measurability is the weakest large cardinal property requiring extenders not equivalent to normal ultrafilters (Section 1.6).

So Magidor forcing is the optimal-strength route to $\mathrm{cf}(\kappa )=\delta >\omega$: nothing beyond the canonical inner-model content of "$o(\kappa )=\delta$" is used.

6.5 Optimality of the hypotheses

Remark 6.5.1 (stated without proof).

The conclusion of Theorem 6.3.1 is optimal in two senses.

1. Bounded subsets are unavoidable over the core model. Mitchell [45] showed: if the ground model is the core model $K$, then any cardinal-preserving extension changing the cofinality of $\kappa$ to an uncountable $\delta <\kappa$ must add new bounded subsets of $\kappa$. Thus the combination “cardinal preservation + no new bounded subsets + $\mathrm{cf}(\kappa )=\delta >\omega$” necessarily requires a prepared ground model: such models were constructed by Mitchell [44] using inner model techniques, and by a pure forcing construction of Gitik [13].
2. The assumption $o(\kappa )=\delta$ is necessary. By Mitchell’s analysis of the core model for sequences of measures [43], changing the cofinality of $\kappa$ to $\delta$ while preserving cardinals requires $o^K(\kappa )\ge \delta$ in the core model. Hence Magidor’s hypothesis is exactly the consistency strength of the conclusion.

Compare the discussion after Theorem 3.2.5: the Radin club itself adds bounded subsets of $\kappa$ below every pair coordinate, in line with Mitchell’s theorem — avoiding them is a matter of preparing the ground model, not of choosing a cleverer forcing.

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Notes

The cone presentation of Section 6.1 is Gitik’s, from the paragraph on p. 1419 of the Handbook chapter: “If $\delta <\kappa$, then $⟨U(\kappa ,\alpha )\mid \alpha <\delta ⟩$ can be split… ${\mathbb{P}}_{\vec{U}}$, above the condition $⟨⟨\kappa ,{\bigcup }_{\alpha <\delta }{Y}_{\alpha }⟩⟩$ is then the Magidor forcing for changing cofinality of $\kappa$ to $\mathrm{cf}(\delta )$.” Lemma 6.1.2 makes explicit the refinement implicit in that identification. Section 6.2 presents Magidor’s original definition from [37]; our notation follows the restatement in Fuchs’ paper (where the forcing is defined with an additional parameter $\tilde{\alpha }$, the “forcing above $\tilde{\alpha }$”, and $\mathbb{M}(\vec{U},\vec{f})=\mathbb{M}(\vec{U},\vec{f},\alpha )$ in his notation). Lemma 6.2.1 is Magidor’s observation that the sets $A_{\gamma },B_{\gamma }$ are of measure one; Lemma 6.2.5 records the standard properties of the Magidor sequence. Theorem 6.3.1 is the main theorem of [37]; the inheritance proof given here is available because of the coherent-sequence approach of Mitchell [42, 43] presented in Chapter 5. Remark 6.5.1 summarizes Mitchell [44, 45] and Gitik [13], as cited in Gitik’s discussion following Theorem 5.12 of the Handbook chapter.

References

Main reference:

• Moti Gitik. Prikry-type forcings. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of Set Theory, pages 1351–1447. Springer, Dordrecht, 2010.

Numbering below follows the bibliography of Gitik’s chapter:

• [13] Moti Gitik. Changing cofinalities and the nonstationary ideal. Israel Journal of Mathematics, 56(3):280–314, 1986.
• [37] Menachem Magidor. Changing cofinality of cardinals. Fundamenta Mathematicae, 99(1):61–71, 1978.
• [42] William J. Mitchell. How weak is a closed unbounded ultrafilter? In Logic Colloquium ‘80 (Prague, 1980), volume 108 of Studies in Logic and the Foundations of Mathematics, pages 209–230. North-Holland, Amsterdam, 1982.
• [43] William J. Mitchell. The core model for sequences of measures. I. Mathematical Proceedings of the Cambridge Philosophical Society, 95(2):229–260, 1984.
• [44] William J. Mitchell. Indiscernibles, skies, and ideals. In Axiomatic Set Theory (Boulder, Colo., 1983), volume 31 of Contemporary Mathematics, pages 161–182. American Mathematical Society, Providence, 1984.
• [45] William J. Mitchell. Applications of the covering lemma for sequences of measures. Transactions of the American Mathematical Society, 299(1):41–58, 1987.

Additional reference for the presentation of Section 6.2:

• Gunar Fuchs. On sequences generic in the sense of Magidor. The Journal of Symbolic Logic, 79(4):1286–1314, 2014.