Friedman-Magidor Theorem for Extenders

• The Friedman-Magidor theorem for extenders is a forcing framework that classifies the number and structure of extender lifts under strong cardinal assumptions.
• It employs nonstationary support products and extender-based Magidor–Radin forcing to produce controlled club and stationary patterns in cardinal arithmetic and cofinalities.
• The theorem generalizes classical measure-counting results, providing insights into embedding lifts, ultrapower equivalences, and optimal consistency strength in set theory.

The Friedman–Magidor theorem for extenders denotes a family of extender analogues of the classical Friedman–Magidor phenomenon. In one formulation, it is a forcing theorem about the number and structure of lifts of ground-model extenders: after forcing with a nonstationary-support product $P^\tau$, every ground-model $(\kappa,\lambda)$-extender $E$ satisfying a generator bound has exactly $j_E(\tau)(\kappa)$ lifts $E^*_\eta$, every relevant extender in the extension arises, up to equivalence, as such a lift, and all lifts of a fixed $E$ have the same ultrapower (Kaplan, 28 Jul 2025). In a second formulation, extender-based Magidor–Radin forcings produce the characteristic Friedman–Magidor pattern of a club or stationary family of cardinals whose cofinalities and power functions are tightly controlled, including versions without top extenders and supercompact-type versions (Gitik et al., 2023).

1. Classical background and extender reformulations

The classical Friedman–Magidor theorem is described in two distinct but related ways in the recent literature. One line, emphasized in the measure-counting context, starts from the statement that if $V=L[U]$ is the minimal inner model with a measurable $\kappa$, and $\tau\le \kappa^{++}$, then there is a cardinal-preserving forcing extension where $\kappa$ is still measurable and carries exactly $(\kappa,\lambda)$0 normal measures. The 2025 reformulation replaces the fine-structural setting by an arbitrary model with a measurable cardinal, avoids forcing over a canonical inner model, avoids self-coding and generalized Sacks forcing, and uses only a simple nonstationary support product; the same paper states that the technique generalizes to a version of the Friedman–Magidor theorem for extenders (Kaplan, 28 Jul 2025).

A second line, emphasized in extender-based Magidor–Radin forcing, treats the Friedman–Magidor phenomenon as the production of a club or stationary pattern of singularization and cardinal arithmetic below an inaccessible cardinal. In its prototypical form, starting from a sufficiently strong large cardinal, one obtains an inaccessible or singular $(\kappa,\lambda)$1 such that SCH fails on a club of singular cardinals below $(\kappa,\lambda)$2, typically with

$(\kappa,\lambda)$3

while outside that club SCH holds or behaves predictably. Later generalizations replace measures by extenders, thereby yielding extender-based Magidor–Radin analogues with stronger patterns and higher consistency strength (Gitik et al., 2023).

This suggests two standard uses of the phrase. In the first, the theorem concerns counting and classifying extender lifts in a forcing extension. In the second, it concerns extender-based Prikry/Magidor/Radin forcing that produces Friedman–Magidor-type club or stationary behavior for cofinalities and power functions.

2. The direct forcing theorem for lifts of extenders

The most explicit theorem carrying the name “Friedman–Magidor for extenders” is Theorem 5.1 of “The number of normal measures, revisited” (Kaplan, 28 Jul 2025). Assume GCH and let $(\kappa,\lambda)$4 satisfy, for every $(\kappa,\lambda)$5,

$(\kappa,\lambda)$6

with both $(\kappa,\lambda)$7 and $(\kappa,\lambda)$8 strictly below the least inaccessible above $(\kappa,\lambda)$9. Let $E$0 be the set of inaccessibles below $E$1, and define

$E$2

ordered by inclusion.

The theorem states that if $E$3 is generic over $E$4, then for every $E$5-extender $E$6 in $E$7 whose generators are all below $E$8, and for each $E$9, there exists a $j_E(\tau)(\kappa)$0-extender

$j_E(\tau)(\kappa)$1

such that

$j_E(\tau)(\kappa)$2

All extenders $j_E(\tau)(\kappa)$3 for $j_E(\tau)(\kappa)$4 have the same ultrapower, meaning that $j_E(\tau)(\kappa)$5 is independent of $j_E(\tau)(\kappa)$6. Conversely, every $j_E(\tau)(\kappa)$7-extender $j_E(\tau)(\kappa)$8 whose generators are all below $j_E(\tau)(\kappa)$9 is, up to equivalence, of the form $E^*_\eta$0 for some ground-model $E^*_\eta$1-extender $E^*_\eta$2 and some $E^*_\eta$3 (Kaplan, 28 Jul 2025).

The special case most directly tied to strong cardinals is Theorem 1.5: if GCH holds and $E^*_\eta$4 is a $E^*_\eta$5-strong cardinal, then for every $E^*_\eta$6 there is a cardinal-preserving forcing extension in which every $E^*_\eta$7-extender $E^*_\eta$8 witnessing that $E^*_\eta$9 is $E$0-strong lifts to exactly $E$1 nonequivalent $E$2-extenders of the extension, all of which have the same ultrapower and again witness $E$3-strongness. Every such extender in the extension arises as such a lift (Kaplan, 28 Jul 2025).

The theorem is “non–fine-structural” in the paper’s terminology. It does not assume that the ground model is $E$4 or $E$5, and it does not use an analysis of canonical inner models. Its content is instead purely forcing-theoretic: it prescribes the number of lifts of each relevant extender and identifies all such extenders in the extension.

3. Forcing mechanism, generic lifts, and the same-ultrapower phenomenon

The underlying forcing is a nonstationary support product. Conditions have nowhere stationary support, and the forcing-theoretic backbone is a fusion lemma for NS-support products. Under GCH, these products preserve all cardinals, and the same framework preserves $E$6 together with the measurability or strongness given by the relevant ground-model extender (Kaplan, 28 Jul 2025).

For a fixed $E$7-extender $E$8 with embedding $E$9, the lifted generic corresponding to $V=L[U]$0 is defined by

$V=L[U]$1

The paper proves that $V=L[U]$2 is $V=L[U]$3-generic over $V=L[U]$4, contains $V=L[U]$5, and yields a lift

$V=L[U]$6

The extender $V=L[U]$7 is then derived from $V=L[U]$8 using the same length $V=L[U]$9 and critical point $\kappa$0 (Kaplan, 28 Jul 2025).

The “same ultrapower” clause is a central feature. For a fixed ground-model extender $\kappa$1, changing $\kappa$2 changes only the atomic choice at the coordinate $\kappa$3 inside $\kappa$4; the rest of the generic filter is unchanged. Because the forcing is a product rather than an iteration, this coordinate change does not alter the transitive collapse of the resulting ultrapower. The theorem therefore gives genuinely different extenders with isomorphic ultrapowers (Kaplan, 28 Jul 2025).

The converse classification depends on Hamkins’ Gap Forcing Theorem. The forcing is factored as $\kappa$5 with a gap below $\kappa$6, and if

$\kappa$7

is induced by an extender in the extension, then $\kappa$8 is definable in $\kappa$9, yielding a ground-model extender $\tau\le \kappa^{++}$0 with $\tau\le \kappa^{++}$1 and $\tau\le \kappa^{++}$2. A further combinatorial argument identifies $\tau\le \kappa^{++}$3 with some $\tau\le \kappa^{++}$4, showing that $\tau\le \kappa^{++}$5 is equivalent to $\tau\le \kappa^{++}$6 (Kaplan, 28 Jul 2025).

The theorem is accompanied by an optimality bound. Goldberg’s Theorem 5.4 shows that, in a GCH context, even if there could be $\tau\le \kappa^{++}$7 extenders with critical point $\tau\le \kappa^{++}$8, at most $\tau\le \kappa^{++}$9 may have the same ultrapower. The special case of $\kappa$0-extenders therefore matches the maximal size of a same-ultrapower family allowed by the paper’s argument (Kaplan, 28 Jul 2025).

4. Extender-based Magidor–Radin forcing without top extenders

A different extender realization of the Friedman–Magidor phenomenon is developed in “Extender-based Magidor-Radin forcings without top extenders” (Gitik et al., 2023). The basic setup assumes that $\kappa$1 is strongly inaccessible, GCH holds in the ground model, and for each inaccessible $\kappa$2 there is a coherent sequence

$\kappa$3

where each $\kappa$4 is an $\kappa$5-extender and

$\kappa$6

The decisive novelty is that there is no $\kappa$7: all extenders live strictly below $\kappa$8, and the target inaccessible is affected only as the limit of the lower hierarchy (Gitik et al., 2023).

For each inaccessible $\kappa$9, the forcing $(\kappa,\lambda)$00 is defined recursively. Its Cohen component is

$(\kappa,\lambda)$01

which is isomorphic to $(\kappa,\lambda)$02. When $(\kappa,\lambda)$03, the forcing also includes an extender-tree component built from the system of ultrafilters $(\kappa,\lambda)$04 on the object spaces $(\kappa,\lambda)$05; the associated $(\kappa,\lambda)$06-$(\kappa,\lambda)$07-trees provide the Magidor–Radin backbone (Gitik et al., 2023).

The induction scheme establishes that each $(\kappa,\lambda)$08 is of size $(\kappa,\lambda)$09, is $(\kappa,\lambda)$10-c.c., and $(\kappa,\lambda)$11 has the Prikry property. It defines a set $(\kappa,\lambda)$12 such that, if $(\kappa,\lambda)$13, $(\kappa,\lambda)$14 is club, while if $(\kappa,\lambda)$15, $(\kappa,\lambda)$16 is bounded. Cardinal arithmetic is determined by the limit points of $(\kappa,\lambda)$17: for $(\kappa,\lambda)$18, either $(\kappa,\lambda)$19 or $(\kappa,\lambda)$20, and

$(\kappa,\lambda)$21

Singularization of regular $(\kappa,\lambda)$22 occurs exactly at the same points. Quotient forcings $(\kappa,\lambda)$23 factor densely over $(\kappa,\lambda)$24 and retain both closure under $(\kappa,\lambda)$25 and the Prikry property (Gitik et al., 2023).

The global forcing is

$(\kappa,\lambda)$26

The deciding theorem shows that if $(\kappa,\lambda)$27 is a $(\kappa,\lambda)$28-name for a function $(\kappa,\lambda)$29 with $(\kappa,\lambda)$30, then $(\kappa,\lambda)$31 is already decided in some $(\kappa,\lambda)$32 with $(\kappa,\lambda)$33. Consequently $(\kappa,\lambda)$34 does not add new functions or sets at $(\kappa,\lambda)$35, and $(\kappa,\lambda)$36 remains inaccessible (Gitik et al., 2023).

The main conclusion is Theorem 8.7: in $(\kappa,\lambda)$37, there is a club $(\kappa,\lambda)$38 such that

$(\kappa,\lambda)$39

Thus SCH fails on a club of cardinals below $(\kappa,\lambda)$40. Section 9 generalizes the Cohen component from $(\kappa,\lambda)$41 to $(\kappa,\lambda)$42, producing stationary classes $(\kappa,\lambda)$43 with

$(\kappa,\lambda)$44

The paper emphasizes an advantage of the construction: fewer cardinals and cofinalities are affected by the forcing (Gitik et al., 2023).

5. Supercompact-type extenders and global Friedman–Magidor patterns

“Supercompact Extender Based Magidor-Radin Forcing” extends the extender-based Magidor–Radin method from short extenders to supercompact-type extenders (Merimovich, 2016). The ground object is a Mitchell increasing sequence

$(\kappa,\lambda)$45

of extenders with common critical point and common directedness degree $(\kappa,\lambda)$46. Extender sequences are pairs $(\kappa,\lambda)$47, ordered by their first coordinates, and the forcing $(\kappa,\lambda)$48 is built from one-step conditions $(\kappa,\lambda)$49 and a global Magidor–Radin threading along increasing critical points (Merimovich, 2016).

The paper’s Main Theorem gives a global Friedman–Magidor statement. In the generic extension $(\kappa,\lambda)$50, there is a set $(\kappa,\lambda)$51 such that $(\kappa,\lambda)$52 is increasing, and for each $(\kappa,\lambda)$53 with $(\kappa,\lambda)$54, the set of earlier critical points is club below $(\kappa,\lambda)$55. The critical point $(\kappa,\lambda)$56 and the directedness $(\kappa,\lambda)$57 are preserved, and the cofinality of $(\kappa,\lambda)$58 is determined by $(\kappa,\lambda)$59: if $(\kappa,\lambda)$60 is $(\kappa,\lambda)$61-regular, then

$(\kappa,\lambda)$62

if $(\kappa,\lambda)$63 and $(\kappa,\lambda)$64, then $(\kappa,\lambda)$65; if $(\kappa,\lambda)$66, then $(\kappa,\lambda)$67 is regular (Merimovich, 2016).

The theorem also includes large-cardinal and continuum conclusions. If $(\kappa,\lambda)$68, then $(\kappa,\lambda)$69 is measurable in the extension. At each such point,

$(\kappa,\lambda)$70

Repeat points of the extender sequence are the mechanism behind measurability: if $(\kappa,\lambda)$71 is a repeat point, then $(\kappa,\lambda)$72 is measurable in the extension, and $(\kappa,\lambda)$73 guarantees the existence of such a repeat point (Merimovich, 2016).

This is the supercompact-extender analogue of the Friedman–Magidor menu. A single forcing produces a closed unbounded family of critical points, and at each point the cofinality, measurability, and continuum are computed from the local parameters $(\kappa,\lambda)$74 and $(\kappa,\lambda)$75. The paper’s examples explicitly exhibit cases where $(\kappa,\lambda)$76, $(\kappa,\lambda)$77 is collapsed, and for limit points $(\kappa,\lambda)$78 of the generic club one has $(\kappa,\lambda)$79 or, more precisely, $(\kappa,\lambda)$80 after successor collapses (Merimovich, 2016).

6. Related extender phenomena and open directions

Two further strands clarify the scope of the extender version of the Friedman–Magidor phenomenon. “Singular cardinals and strong extenders” studies short $(\kappa,\lambda)$81-extenders witnessing that $(\kappa,\lambda)$82 is $(\kappa,\lambda)$83-strong and asks whether the singular cardinal $(\kappa,\lambda)$84 remains singular or becomes inaccessible in the corresponding ultrapower (Apter et al., 2012). The paper introduces the distinction between a good witness, where

$(\kappa,\lambda)$85

and a bad witness, where

$(\kappa,\lambda)$86

If $(\kappa,\lambda)$87 is $(\kappa,\lambda)$88-strong, then every singular $(\kappa,\lambda)$89 with $(\kappa,\lambda)$90 and $(\kappa,\lambda)$91 has a good witness. If $(\kappa,\lambda)$92 is inaccessible and $(\kappa,\lambda)$93 is $(\kappa,\lambda)$94-strong, then there is a club $(\kappa,\lambda)$95 such that every singular $(\kappa,\lambda)$96 has a bad witness. In coherent non-overlapping models $(\kappa,\lambda)$97, the paper classifies all $(\kappa,\lambda)$98-extenders witnessing $(\kappa,\lambda)$99-strongness and proves consistency results in which the unique such extender is good, or alternatively bad (Apter et al., 2012). This refines the singularization aspect of the Friedman–Magidor picture from the perspective of strong extenders.

A different application appears in “On Singular Stationarity II,” which describes its main results as an “extender-based Friedman–Magidor theorem” for tightly stationary sequences (Ben-Neria, 2017). Using short-extenders forcing, the paper proves that if $E$00 is an increasing sequence of $E$01-extendible cardinals, then for every sequence of fixed-cofinality stationary sets $E$02 with $E$03, there is a generic extension in which the sequence is tightly stationary. A second theorem lowers the large-cardinal strength to a sequence of $E$04-strong cardinals and obtains an analogous result on a subsequence $E$05 of the $E$06’s (Ben-Neria, 2017). Here the extender mechanism is not a counting theorem for lifts, but a forcing construction that adds scales and continuity points with the global stationarity properties characteristic of the Foreman–Magidor framework.

Several open problems remain explicit. The 2025 paper asks whether it is consistent, with GCH, that there are $E$07 normal measures on $E$08 but only a single normal ultrapower; whether a Friedman–Magidor-type analysis can be combined with violation of GCH at a measurable $E$09 in an arbitrary model with a $E$10-strong cardinal, without relying on inner model theory; and whether there is a Friedman–Magidor theorem for fine, normal ultrafilters on $E$11 for $E$12 (Kaplan, 28 Jul 2025). The 2012 paper asks for the exact consistency strength of the existence of a bad witness and whether it is consistent that there are exactly two $E$13-extenders witnessing that $E$14 is $E$15-strong, one good and the other bad (Apter et al., 2012).

Taken together, these results show that “Friedman–Magidor theorem for extenders” does not designate a single isolated statement. It names a stable extender-theoretic pattern: forcing or ultrapower constructions that replace measures by extenders, preserve a strong degree of structural control, and then classify either the lifts of extenders themselves or the club and stationary configurations of cofinalities, power functions, and stationarity principles that those extenders make possible.