Friedman–Magidor Theorem: Lifting Normal Measures

• The Friedman–Magidor theorem is a combinatorial result showing that forcing with P^τ produces exactly τ distinct lifts of any ground-model normal measure on a measurable cardinal.
• It employs nonstationary-support product forcing to ensure that every normal measure in the extension uniquely corresponds to a lifted ground-model measure.
• The coding-free approach avoids canonical inner models and fine-structural tools, and it generalizes to extenders, broadening applications to larger cardinal frameworks.

Searching arXiv for the cited paper and closely related sources. The Friedman–Magidor theorem concerns the possible number of normal measures carried by a measurable cardinal after forcing. In the formulation presented in Kaplan’s “The number of normal measures, revisited” (Kaplan, 28 Jul 2025), for every measurable cardinal $\kappa$ and every ordinal $\tau \le \kappa^{++}$, there is a cardinal-preserving forcing extension in which each ground-model normal measure on $\kappa$ has exactly $\tau$ distinct lifts, every normal measure in the extension is such a lift, and therefore $\kappa$ carries exactly $\tau$ normal measures. The result is a new “coding-free” version of the Friedman–Magidor theorem, differing from the original approach by avoiding forcing over canonical inner models, fine-structural tools, and self-coding machinery, while using only nonstationary support product forcing (Kaplan, 28 Jul 2025).

1. Formal statement and scope

Kaplan’s main theorem is stated for an arbitrary measurable cardinal $\kappa$ and an arbitrary ordinal $\tau \le \kappa^{++}$ (Kaplan, 28 Jul 2025). There is a forcing $P^\tau \in V$ that preserves the measurability of $\kappa$, and for any $\tau \le \kappa^{++}$0-generic filter $\tau \le \kappa^{++}$1, two conclusions hold. First, every normal measure $\tau \le \kappa^{++}$2 on $\tau \le \kappa^{++}$3 admits exactly $\tau \le \kappa^{++}$4 distinct lifts

$\tau \le \kappa^{++}$5

to normal measures on $\tau \le \kappa^{++}$6 in $\tau \le \kappa^{++}$7. Second, every normal measure on $\tau \le \kappa^{++}$8 in $\tau \le \kappa^{++}$9 is one of these lifts. In particular, in the extension $\kappa$0, the cardinal $\kappa$1 carries exactly $\kappa$2 normal measures (Kaplan, 28 Jul 2025).

A further feature appears when $\kappa$3. In that case, all lifts of a fixed ground-model normal measure $\kappa$4 have the same ultrapower. The theorem therefore distinguishes two layers of multiplicity: the number of distinct lifted measures, and the structure of the corresponding ultrapower models. The latter collapses to a single ultrapower for each $\kappa$5 in the range $\kappa$6 (Kaplan, 28 Jul 2025).

This formulation is presented as a new version of the Friedman–Magidor theorem. Its novelty, as explicitly emphasized, lies not only in the enumeration of lifts but also in the method: it does not rely on canonical inner models or fine structure, and it generalizes to extenders (Kaplan, 28 Jul 2025).

2. Normal measures, ultrapowers, and lifts

A normal measure on a measurable cardinal $\kappa$7 is defined as a filter $\kappa$8 such that $\kappa$9 is an ultrafilter, $\tau$0 is $\tau$1-complete, and whenever $\tau$2 is regressive on a $\tau$3-large set $\tau$4, there is some $\tau$5 with $\tau$6 (Kaplan, 28 Jul 2025). Equivalently, such a measure induces an ultrapower embedding

$\tau$7

with critical point $\tau$8 (Kaplan, 28 Jul 2025).

The theorem is formulated in terms of lifts of normal measures across forcing extensions. If $\tau$9 is a normal measure on $\kappa$0 and $\kappa$1 is $\kappa$2-generic, then a lift of $\kappa$3 to $\kappa$4 is a normal measure $\kappa$5 on $\kappa$6 such that the embedding $\kappa$7 extends to an embedding

$\kappa$8

where $\kappa$9 is generic over $\tau$0 and $\tau$1 (Kaplan, 28 Jul 2025).

This definition isolates the precise sense in which the extension does not create unrelated measure structure. A normal measure in the extension is relevant only insofar as it extends a ground-model embedding, and the theorem’s second clause asserts that every normal measure in the forcing extension arises this way. A plausible implication is that the result is not merely a counting theorem for ultrafilters, but a structural classification theorem for the normal measures on $\tau$2 after forcing.

3. The splitting forcing $\tau$3

The forcing used in the coding-free theorem is the splitting forcing $\tau$4, described as a nonstationary-support product forcing (Kaplan, 28 Jul 2025). Let $\tau$5 be the class of inaccessible cardinals below $\tau$6. One fixes a “canonical” function

$\tau$7

where $\tau$8 indexes $\tau$9 many values (Kaplan, 28 Jul 2025).

Conditions in $\kappa$0 are partial functions

$\kappa$1

such that $\kappa$2 is a nowhere stationary set of inaccessibles, and for each $\kappa$3, one has $\kappa$4 (Kaplan, 28 Jul 2025). The order is by extension: $\kappa$5 Equivalently, $\kappa$6 is the product

$\kappa$7

with nonstationary support, where

$\kappa$8

is the one-step “atomic” forcing that either does nothing or picks a value in $\kappa$9 (Kaplan, 28 Jul 2025).

The key forcing-theoretic facts stated for $\tau \le \kappa^{++}$0 are concise and central. A fusion/factorization argument, identified as the Fusion Lemma for NS-support products, shows that $\tau \le \kappa^{++}$1 preserves $\tau \le \kappa^{++}$2 and $\tau \le \kappa^{++}$3; under GCH it preserves all cardinals (Kaplan, 28 Jul 2025). If $\tau \le \kappa^{++}$4 is generic, then it induces a partition of $\tau \le \kappa^{++}$5 into $\tau \le \kappa^{++}$6 many pairwise disjoint stationary sets

$\tau \le \kappa^{++}$7

(Kaplan, 28 Jul 2025). Moreover, each coordinate $\tau \le \kappa^{++}$8 can be recovered from the stationary set $\tau \le \kappa^{++}$9, so $P^\tau \in V$0 many distinct generics arise (Kaplan, 28 Jul 2025).

This forcing description is one of the theorem’s most distinctive features. The data explicitly contrasts it with the more elaborate machinery of the original theorem: the new proof relies only on nonstationary support product forcing, not on interleaved coding or fine-structural self-coding posets (Kaplan, 28 Jul 2025).

4. Mechanism of the proof

The proof has two principal components: producing exactly $P^\tau \in V$1 lifts from each ground-model normal measure, and showing that no additional normal measures appear in the extension (Kaplan, 28 Jul 2025).

For the first component, one fixes a normal measure $P^\tau \in V$2 on $P^\tau \in V$3 and, for each $P^\tau \in V$4, defines a filter $P^\tau \in V$5 over $P^\tau \in V$6 by

$P^\tau \in V$7

A standard fusion-in-the-ultrapower argument shows that each $P^\tau \in V$8 meets every dense open subset of $P^\tau \in V$9 (Kaplan, 28 Jul 2025). By Silver’s criterion, the ultrapower embedding lifts to

$\kappa$0

and $\kappa$1 is the normal measure derived from this lifted embedding (Kaplan, 28 Jul 2025). Distinct values of $\kappa$2 produce distinct measures, so each $\kappa$3 yields exactly $\kappa$4 lifts. The data further states that because only the coordinate at $\kappa$5 is altered, all these lifts share the same ultrapower model $\kappa$6 (Kaplan, 28 Jul 2025). In the abstract and theorem statement, this identical-ultrapower conclusion is recorded specifically for the case $\kappa$7 (Kaplan, 28 Jul 2025).

For the second component, suppose $\kappa$8 is any normal measure on $\kappa$9. By Hamkins’ Gap Forcing Theorem, the restriction $\tau \le \kappa^{++}$00 is a normal measure $\tau \le \kappa^{++}$01 (Kaplan, 28 Jul 2025). If

$\tau \le \kappa^{++}$02

is the associated ultrapower embedding, then its critical point remains $\tau \le \kappa^{++}$03, and $\tau \le \kappa^{++}$04 (Kaplan, 28 Jul 2025). A routine check yields

$\tau \le \kappa^{++}$05

Thus every normal measure in the extension is one of the previously constructed lifts (Kaplan, 28 Jul 2025).

The overall argument therefore has an exactness property on both sides: there are at least $\tau \le \kappa^{++}$06 lifts because one can force $\tau \le \kappa^{++}$07 at the top coordinate, and there are at most $\tau \le \kappa^{++}$08 lifts because every normal measure in the extension is determined by its ground-model restriction together with that same top-coordinate value. This suggests that the stationary splitting performed by $\tau \le \kappa^{++}$09 is not incidental but the combinatorial device that rigidly controls the lift spectrum.

5. Relation to the original Friedman–Magidor theorem

The comparison with the original Friedman–Magidor approach is explicit. The original theorem, identified as Friedman–Magidor 2009, used forcing over the canonical inner model $\tau \le \kappa^{++}$10, fine-structural self-coding posets, generalized Sacks forcing, and nonstationary-support iterations with interleaved coding (Kaplan, 28 Jul 2025). By contrast, Kaplan’s version forces directly over $\tau \le \kappa^{++}$11 with the simple NS-support product $\tau \le \kappa^{++}$12, and does so without inner models, without fine structure, and without self-coding (Kaplan, 28 Jul 2025).

The significance of that contrast is also stated directly. The new method can be applied “in the field of large cardinals beyond the current reach of the inner model program,” and the details specifically mention levels such as supercompact and strongly compact (Kaplan, 28 Jul 2025). The theorem’s proof technology is therefore positioned as structurally independent of the canonical-inner-model framework that shaped the original argument.

Another point of contrast concerns the lifted ultrapowers. In the new approach, when $\tau \le \kappa^{++}$13, all $\tau \le \kappa^{++}$14 lifts of a given normal measure have the same ultrapower (Kaplan, 28 Jul 2025). That feature is singled out as one of the notable ways in which the revised theorem differs from the original Friedman–Magidor theorem. A plausible implication is that the new forcing separates multiplicity of lifted measures from multiplicity of resulting ultrapower models more cleanly than the earlier coding-based construction.

6. Generalization to extenders

The method extends from normal measures to $\tau \le \kappa^{++}$15-extenders (Kaplan, 28 Jul 2025). In the stated generalization, one assumes GCH and that $\tau \le \kappa^{++}$16 is $\tau \le \kappa^{++}$17-strong in $\tau \le \kappa^{++}$18 (Kaplan, 28 Jul 2025). For each $\tau \le \kappa^{++}$19, the same splitting forcing $\tau \le \kappa^{++}$20 yields an extension in which every ground-model $\tau \le \kappa^{++}$21-extender $\tau \le \kappa^{++}$22 whose generators lie below $\tau \le \kappa^{++}$23 admits exactly $\tau \le \kappa^{++}$24 many lifts

$\tau \le \kappa^{++}$25

all of which induce the same ultrapower (Kaplan, 28 Jul 2025). Conversely, every $\tau \le \kappa^{++}$26-extender in $\tau \le \kappa^{++}$27 whose generators lie below $\tau \le \kappa^{++}$28 is one of these lifts (Kaplan, 28 Jul 2025).

The proof is said to proceed exactly as for measures, using the generalized Fusion Lemma and Hamkins’ Gap argument to lift the extender embedding $\tau \le \kappa^{++}$29 at the top coordinate $\tau \le \kappa^{++}$30 (Kaplan, 28 Jul 2025). The central pattern of the measure case is therefore preserved: one obtains controlled multiplicity by splitting at the top coordinate, and one proves exhaustiveness by showing that every new object in the extension restricts to a ground-model object.

This extender version clarifies the broader methodological content of the theorem. The result is not limited to ultrafilters on $\tau \le \kappa^{++}$31; rather, it exemplifies a general forcing pattern for calibrating the number of lifted large-cardinal objects while retaining control over the associated ultrapowers. Within the limits of the stated hypotheses, the theorem therefore occupies a place at the interface of forcing, measurability, and extender-based large-cardinal embeddings (Kaplan, 28 Jul 2025).