Pith Number

pith:6PSNABKW

pith:2026:6PSNABKWXVE6LMFPJX5PS4D7CV

not attested not anchored not stored refs pending

Strong Completeness of Provability Logic for Uncountable Languages

Grigorii Stepanov, Mohammad Golshani, Reihane Zoghifard

Provability logic GL fails strong completeness for modal languages of cardinality (2^|λ| + ℵ₀)^+ on ordinals with generalized Icard topologies.

arxiv:2602.09470 v2 · 2026-02-10 · math.LO

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{6PSNABKWXVE6LMFPJX5PS4D7CV}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

For an ordinal λ>0, there exists a GL-consistent set of formulas having neither (Θ, I_λ)-model nor (Θ, τ_c +λ)-model when the language has cardinality (2^|λ|+ℵ₀)^+; λ-bouquet spaces yield strong completeness of GL for languages of cardinality λ.

C2weakest assumption

The generalized Icard topologies I_λ and τ_c +λ are the right semantics for testing strong completeness; if other topologies or different model classes are intended, the failure result may not apply.

C3one line summary

Strong completeness of GL fails for modal languages of size (2^|λ|+ℵ₀)^+ on generalized Icard topologies but holds for GL and GL.3 on λ-bouquet and ultralinear λ-bouquet spaces.

Receipt and verification

First computed	2026-05-18T02:44:31.240437Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f3e4d00556bd49e5b0af4dfaf9707f154715527a008ca2dd9157ce9fa64fab33

Aliases

arxiv: 2602.09470 · arxiv_version: 2602.09470v2 · doi: 10.48550/arxiv.2602.09470 · pith_short_12: 6PSNABKWXVE6 · pith_short_16: 6PSNABKWXVE6LMFP · pith_short_8: 6PSNABKW

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/6PSNABKWXVE6LMFPJX5PS4D7CV \
  | 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: f3e4d00556bd49e5b0af4dfaf9707f154715527a008ca2dd9157ce9fa64fab33

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "80100e725041ad136e6fbd49a7baa6d9a254b11da08f2fae1613c0e411dd5b6f",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.LO",
    "submitted_at": "2026-02-10T07:06:59Z",
    "title_canon_sha256": "3fe8ca01a7d565c8610b9c4a84464611db60423395364c8174226f7a6b044a46"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2602.09470",
    "kind": "arxiv",
    "version": 2
  }
}