Completeness Senses and the Levi-Civita Field: Ordered Non-Archimedean Number Systems Beyond Ostrowski's Classification

In the paper's words

Ostrowski's theorem classifies every nontrivial rank-1 absolute value on the rationals; this paper distinguishes three senses of completeness and exhibits the Levi-Civita field as a Cauchy-complete, ordered, real-closed, non-Archimedean counterexample.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. Ostrowski's theorem (1916) classifies every nontrivial absolute value on the rational numbers: it is equivalent either to the Archimedean absolute value or to a p-adic absolute value for some prime p.

   from 1. The scope of Ostrowski's theorem

2. Every nonempty set with an upper bound has a least upper bound.

   from 2. Three senses of completeness

3. The treatment of positional notation as an ultrametric tree and the adelic program's use of Ostrowski's theorem provide the surrounding structure in which this refinement sits.

   from 7. Relation to prior work

What would falsify it

1. One member of that family — the Levi-Civita field — is ordered, complete in a metric sense, real-closed, and non-Archimedean, and therefore serves as a concrete counterexample to the claim that the reals are the unique complete ordered field.

   from Completeness Senses and the Levi-Civita Field: Ordered Non-Archimedean Number Systems Beyond Ostrowski's Classification

2. The Levi-Civita field is therefore a genuine counterexample to the unqualified claim that the reals are the unique complete ordered field.

   from 3. The Levi-Civita field

3. The apparent counterexample appears only when the decimal is given nonstandard indices .

   from 4. The decimal identity and nonstandard indices

Where the claim stops

1. On the axis of Cauchy completeness they are not: the Levi-Civita field is ordered, complete, real-closed, and non-Archimedean, carrying infinitesimals the reals cannot admit.

   from 8. Conclusion

2. Once the senses are separated, a family of ordered number systems emerges that the standard classification does not name.

   from Completeness Senses and the Levi-Civita Field: Ordered Non-Archimedean Number Systems Beyond Ostrowski's Classification

3. It does not hold for the hyperreals or the surreals.

   from 2. Three senses of completeness

Where to read

1. 1. The scope of Ostrowski's theorem143 words · 1 min
2. 2. Three senses of completeness234 words · 1 min
3. 3. The Levi-Civita field142 words · 1 min
4. 4. The decimal identity and nonstandard indices147 words · 1 min
5. 5. Higher-rank valuations84 words · 1 min
6. 6. Computational verification218 words · 1 min
7. 7. Relation to prior work124 words · 1 min
8. 8. Conclusion120 words · 1 min