Numbers and Computability
Arithmetic and Algorithmic Foundations
Summary
This work establishes the computational foundations of the Universal Language. Natural numbers
are constructed as iterated point-translations, integers and rationals via directed segments and ratios.
The parsing algorithm is proven to run in O(n log n) time. Most significantly, the UL is shown to encode
Robinson's Q (all 7 axioms verified), making it subject to Gödel incompleteness — a profound result
confirming that the UL has the expressive power of arithmetic.
Status Summary
Proven(4)
- •ℕ constructible as iterated translations
- •ℤ and ℚ constructible via geometry
- •O(n log n) parsing algorithm
- •Robinson's Q fully encoded (Gödel applies)
Framework(2)
- •ℝ via geometric Dedekind cuts
- •Description complexity DC_UL defined
Conjectured(3)
- •Topological equivalence undecidable
- •Generation problem is NP-hard
- •Ω(n log n) parsing lower bound
Key Ideas
ℕ as Iterated Translations
ProvenNatural numbers are constructible in UL as iterated point-translations from origin O using unit U.
ℤ and ℚ via Geometry
ProvenIntegers are constructed via directed segments; rationals via segment ratios.
O(n log n) Parsing
ProvenThe UL parsing algorithm runs in O(n log n) time for n primitives, assuming unique parse. Decidable for Euclidean/similarity/affine/projective equivalence in O(n).
Encodes Robinson's Q
ProvenThe UL encodes all 7 axioms of Robinson's Q, verified via explicit coordinate computation. Subject to Gödel incompleteness.
Descriptional Complexity DC_UL
FrameworkA well-defined measure of expression complexity, analogous to Kolmogorov complexity.
Not Yet Addressed
- Transcendental number constructions beyond π and e
- Probability distributions over glyph space
- Complex numbers and higher dimensions
- Parse disambiguation for ambiguous constructions
Prerequisites
- foundations/paradigm.md
- foundations/formal-foundations.md
Related Open Problems
Source Document
frontier/expedition-one/numbers-and-computability.md