The Computable Continuum: Depth Without Breadth

In the paper's words

The uncountability of the real numbers conflates Archimedean completeness (depth) and set-theoretic cardinality explosion (breadth). All physically relevant properties are preserved by computable reals.

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

Numbers it reports

1. The OC criterion excludes: Object Reason for Exclusion Physical Cost Non-computable reals (Chaitin's $\Omega$, etc.) No finite protocol produces them with computable error bound None — they are unfalsifiable (Theorem 4.3) The Continuum Hypothesis Quantifies over physically inaccessible sets None — cannot affect any measurement (Theorem 4.5) Arbitrary Dedekind cuts May require non-computable specification None — the computable cuts are sufficient for all computable suprema The full power set of $\mathbb{N}$ Generates physically equivalent reals None — the excess reals are physical duplicates of computable ones

   from 5.2 What This Excludes

How it is done

1. Non-computable reals are physically unfalsifiable: For any two non-computable reals x, y that agree on all rational approximations with computable error bounds, no finite measurement protocol can produce measurement-distinguishable predictions parameterized by x versus y.

   from 1.2 The Thesis

2. Theorem 2.4 ($\mathbb{R}_c$ is computably complete).

   from 2.2 Completeness and Connectedness

3. Theorem 2.7 (Intermediate Value for $\mathbb{R}_c$).

   from 2.3 Intermediate Value and Real Closure

4. The Specker sequence (Specker 1949) is the canonical example demarcating depth from breadth.

   from 3.1 Construction

What would falsify it

1. The truth value of the Continuum Hypothesis (CH) cannot affect any measurement-distinguishable physical prediction.

   from 4.3 The Continuum Hypothesis Has Zero Physical Content

Where the claim stops

1. Most real numbers are non-computable: Chaitin's $\Omega$, the Specker sequence limit, and continuum-many other reals cannot be approximated by any finite Turing machine with computable error bounds.

   from 1.1 Three Facts, Two Properties

2. The middle term does not imply the conclusion.

   from 1.1 Three Facts, Two Properties

3. Non-computable reals are physically unfalsifiable: For any two non-computable reals x, y that agree on all rational approximations with computable error bounds, no finite measurement protocol can produce measurement-distinguishable predictions parameterized by x versus y.

   from 1.2 The Thesis

Where to read

1. 1. Introduction: The Conflation Problem372 words · 2 min
2. 2. Preliminaries510 words · 2 min
3. 3. The Specker Sequence: The Precise Boundary250 words · 1 min
4. 4. Physical Unfalsifiability of Non-Computable Reals567 words · 2 min
5. 5. Operational Ontology248 words · 1 min
6. 6. Implications for Physics309 words · 1 min
7. 7. Relation to the Trilogy138 words · 1 min