Pith Number

pith:HVNVI46S

pith:2026:HVNVI46SW6XEB2UFOKTGOEQHPR

not attested not anchored not stored refs pending

The reals as a subset of an ultraproduct of finite fields

Roee Sinai

If an ultraproduct of prime finite fields contains the algebraic reals, then that copy or its closure must arise from one of several new constructions using mostly internal sets.

arxiv:2603.07508 v3 · 2026-03-08 · math.LO

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\usepackage{pith}
\pithnumber{HVNVI46SW6XEB2UFOKTGOEQHPR}

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Record completeness

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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

if an ultraproduct of prime finite fields includes a copy of the algebraic real numbers then either this copy or its algebraic closure can be constructed in some of these ways. We also show that no copy of the field of real numbers inside such an ultraproduct can ever be constructed in any of these ways, but there is either a hyperreal field or an algebraically closed field of cardinality larger or equal to the continuum that can be.

C2weakest assumption

The precise definitions of the new construction methods (external subsets built mostly from internal sets) and the exact meaning of 'includes a copy' are load-bearing; if these notions are interpreted differently or if the ultraproduct fails to satisfy the internal-set properties assumed, the positive and negative existence claims collapse.

C3one line summary

In ultraproducts of prime finite fields, algebraic reals admit internal-heavy constructions under stated conditions, but the full real field never does, while hyperreals or continuum-sized algebraically closed fields do.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-22T01:03:59.750224Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3d5b5473d2b7ae40ea8572a66712077c47b2b983c5b848914b6bd10de89f8954

Aliases

arxiv: 2603.07508 · arxiv_version: 2603.07508v3 · doi: 10.48550/arxiv.2603.07508 · pith_short_12: HVNVI46SW6XE · pith_short_16: HVNVI46SW6XEB2UF · pith_short_8: HVNVI46S

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/HVNVI46SW6XEB2UFOKTGOEQHPR \
  | 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: 3d5b5473d2b7ae40ea8572a66712077c47b2b983c5b848914b6bd10de89f8954

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0730f217bcad135a09c3cef24a84a0c66c643357470d45e4a2bf73fb28abd332",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-sa/4.0/",
    "primary_cat": "math.LO",
    "submitted_at": "2026-03-08T07:25:46Z",
    "title_canon_sha256": "5588d8c4cf3b10d02c3e495c91bb7d4d88b2353b1e98d99131fad1787241e23b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.07508",
    "kind": "arxiv",
    "version": 3
  }
}