Pith Number

pith:26EBIPPY

pith:2025:26EBIPPYTKJHGTL253ICQWVBXK

not attested not anchored not stored refs pending

Combinatorics in Higher Solovay Models

Alejandro Poveda, Sebastiano Thei

Assuming the consistency of ZFC with large cardinals, a model exists in which aleph_omega is a strong limit and L(P(aleph_omega)) satisfies the aleph_omega-perfect set property for all subsets of sequences, has no scale, fails SCH and AP,,

arxiv:2509.18991 v4 · 2025-09-23 · math.LO

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\pithnumber{26EBIPPYTKJHGTL253ICQWVBXK}

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Claims

C1strongest claim

Assuming the consistency of ZFC with appropriate large cardinal axioms, there is a model of ZFC in which aleph_omega is a strong limit cardinal and L(P(aleph_omega)) satisfies: every A subset (aleph_omega)^omega has the aleph_omega-PSP, there is no scale at aleph_omega, SCH fails at aleph_omega, AP fails at aleph_omega, and TP holds at aleph_omega+1. This is the first Solovay-type model at the first singular cardinal.

C2weakest assumption

The consistency of ZFC together with the appropriate large cardinal axioms is assumed in order to produce the model; without this background consistency the forcing or inner-model construction that arranges the five listed properties at aleph_omega cannot be carried out.

C3one line summary

From large cardinals, constructs a model where aleph_omega is strong limit, L(P(aleph_omega)) has aleph_omega-PSP, no scales, SCH and AP fail, TP holds at aleph_omega+1, answering Woodin's question on SCH vs AP.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-06-04T01:08:32.198509Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

d788143df89a92734d7aeed0285aa1ba85e4cb641da0677b7ccfafdb55278745

Aliases

arxiv: 2509.18991 · arxiv_version: 2509.18991v4 · doi: 10.48550/arxiv.2509.18991 · pith_short_12: 26EBIPPYTKJH · pith_short_16: 26EBIPPYTKJHGTL2 · pith_short_8: 26EBIPPY

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/26EBIPPYTKJHGTL253ICQWVBXK \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: d788143df89a92734d7aeed0285aa1ba85e4cb641da0677b7ccfafdb55278745

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d2cd93ba4f307c4d8f819894d6a68e4acd54648b7890d6a553d952e20b07ce53",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.LO",
    "submitted_at": "2025-09-23T13:39:14Z",
    "title_canon_sha256": "fdffaf156d9523982648ecc3102e4d789123c772daa4fcbcbfbe509d15d94a46"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2509.18991",
    "kind": "arxiv",
    "version": 4
  }
}