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arxiv: 2412.09683 · v2 · submitted 2024-12-12 · 🧮 math.LO

Applications of the Magidor Iteration to Ultrafilter Theory

Tom Benhamou , Gabriel Goldberg This is my paper

Pith reviewed 2026-05-23 07:02 UTC · model grok-4.3

classification 🧮 math.LO
keywords ultrafiltersPrikry forcingMagidor iterationUltrapower Axiommeasurable cardinalsnormal ultrafiltersultrapowersforcing
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The pith

Sums of normal ultrafilters after the Magidor iteration of Prikry forcings can be characterized, separating the weak Ultrapower Axiom from the Ultrapower Axiom.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterizes sums of normal ultrafilters after the Magidor iteration, which is the product of Prikry forcings over a discrete set of measurable cardinals. This characterization is applied to produce a model in which the weak Ultrapower Axiom holds but the full Ultrapower Axiom fails. The same technique yields a non-rigid ultrapower and two uniform ultrafilters on distinct cardinals that induce the same ultrapower. A sympathetic reader would care because these results distinguish two axioms that concern the structure of ultrapowers in set theory and show how specific forcing iterations interact with ultrafilter properties.

Core claim

We characterize sums of normal ultrafilters after the Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower Axiom. We also construct a non-rigid ultrapower and two uniform ultrafilters on different cardinals that have the same ultrapower.

What carries the argument

The Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals, which preserves normality sufficiently to allow explicit description of ultrafilter sums.

Load-bearing premise

The Magidor iteration preserves normality and other relevant properties of the ultrafilters so that their sums admit a complete characterization.

What would settle it

An explicit sum of normal ultrafilters after the iteration whose structure fails the claimed characterization, or a model where the weak Ultrapower Axiom holds but the full axiom fails without matching the constructed properties.

Figures

Figures reproduced from arXiv: 2412.09683 by Gabriel Goldberg, Tom Benhamou.

Figure 1
Figure 1. Figure 1: The case that m /∈ d Note that by L´evy-Solovay, jD u,x m ↾ Nm[Gm] = j Nm[Gm] D∗m The key point is that (3) j Nm[Gm] D∗m ↾ Nm = jD0m ↾ Nm To see this, let α be the least ordinal such that crit(i Pm α,α+1) > em(δm) and consider the model Nm,α[Gm ↾ δ 0 m]. Note that Gm ↾ δ 0 m is Nα,m-generic for a forcing which has smaller cardinality than the critical point of the embedding i Pm α,∞. So by the L´evy-Solova… view at source ↗
Figure 2
Figure 2. Figure 2: The case that u(m) = 1. The commutativity of the second square from the bottom is proved as in Equation 4. The only other part of the diagram whose commutativity is not immediate is i Pm+1 0,α ◦ jEm = j i Pm 0,α (Em) ◦ i P ′m 0,α where P ′ m = em(VWm+1 ) and i P ′m 0,α : P ′ m → N′ m,α is the α th stage of the complete iteration of P ′ m by em(jWm+1 (U~ )) above κ. N′ m,α Nm+1,α P ′ m Pm+1 i Pm 0,α (Em) i … view at source ↗
Figure 3
Figure 3. Figure 3: The case where u(m) 6= 1 Lemma 5.6. Fix an internal iteration (D0, ..., Dn−1) of normal ultrafilters, let d and d ′ be as in the paragraph following Theorem 5.3. Suppose that d = d ′ and fix u : d → {0, 1} and x : d → ω. Let W¯ = P(D0, ..., Dn−1) u x . Then in V [G] there is an internal iteration (F0, ..., Fℓ−1) of normal ultrafilters such that V [G]W¯ = V [G]F0,...,Fℓ−1 . Proof. An easy induction using th… view at source ↗
Figure 4
Figure 4. Figure 4: The decomposition of jW¯ m+1. We claim that i[Dm] ⊆ U˜. To see this, let X ∈ Dm, then [id]Dm ∈ jDm(X) ⇒ ¯δm ∈ k(jDm (X)) ⇒ kU˜ ([id]U˜ ) ∈ kU˜ (jU¯ (i(X))) ⇒ [id]U˜ ∈ jU¯ (i(X)) ⇒ i(X) ∈ U˜ Next we will prove that U˜ is equal to one of the following ultrafilters: • i(Dm). • pδ α+n m for some n < ω [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
read the original abstract

We characterize sums of normal ultrafilters after the Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower Axiom. We also construct a non-rigid ultrapower and two uniform ultrafilters on different cardinals that have the same ultrapower.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper characterizes sums of normal ultrafilters after the Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals. It applies this characterization to prove that the weak Ultrapower Axiom is not equivalent to the Ultrapower Axiom, and to construct a non-rigid ultrapower together with two uniform ultrafilters on different cardinals that have the same ultrapower.

Significance. If the central characterization holds, the work supplies explicit preservation lemmas for normality and ultrafilter properties under the Magidor iteration, yielding concrete separations between wUA and UA as well as the stated constructions. These results provide falsifiable distinctions in ultrapower theory and strengthen the toolkit for analyzing sums of normal measures after Prikry-type iterations.

minor comments (2)
  1. The abstract refers to 'the Magidor iteration (product)'; a brief clarification in §1 on whether the construction is strictly an iteration or a product would aid readers unfamiliar with the terminology.
  2. Notation for the discrete set of measurable cardinals and the resulting ultrafilter sums could be introduced with a short table or diagram in §2 to improve readability of the preservation lemmas.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of its contributions, and recommendation to accept. The report correctly identifies the central characterization of ultrafilter sums under the Magidor iteration and its applications to separating wUA from UA, as well as the constructions of non-rigid ultrapowers and matching ultrafilters.

Circularity Check

0 steps flagged

Derivation is self-contained; no circular steps identified

full rationale

The paper's central claims rest on preservation lemmas for normality and ultrafilter properties under the Magidor iteration of Prikry forcings, which are established directly from the forcing construction rather than from the target characterization or UA non-equivalence results. These lemmas are independent of the applications to wUA vs UA and the non-rigid ultrapower constructions. No equations, self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The work relies on standard external set-theoretic techniques without reducing the conclusions to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work assumes ZFC plus the existence of a discrete set of measurable cardinals; no free parameters, ad-hoc axioms, or invented entities are mentioned in the abstract.

axioms (2)
  • standard math ZFC set theory
    Standard background for all forcing and ultrafilter arguments in the paper.
  • domain assumption Existence of measurable cardinals
    Required for the Prikry forcings and the ultrafilters being summed.

pith-pipeline@v0.9.0 · 5577 in / 1307 out tokens · 44734 ms · 2026-05-23T07:02:00.264005+00:00 · methodology

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Reference graph

Works this paper leans on

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