Odoardo Volonterio
## Abstract Draw a line. Almost every point on it corresponds to a real number that no finite formula, algorithm, or description specifies uniquely. In this paper we call such real numbers *ghost numbers*. The starting point is an infinite sequence of digits $$ \mathbf{D}=(D_1,D_2,D_3,\ldots), \qquad D_k\in\{0,1,\ldots,b-1\}. $$ The sequence is the primary object. The decimal point is introduced only afterwards, by convention, and the natural choice is to place it after the first digit: $$ G^{(\star)} = D_1.D_2D_3\ldots = \sum_{k=1}^{\infty}\frac{D_k}{b^{k-1}} \in (0,b). $$ Fix, in the metalanguage, a background universe $V\models\mathrm{ZFC}$, and let $\mathcal D$ denote the set of real numbers definable without parameters in $V$. Since $\mathcal D$ is at most countable, every continuous distribution assigns probability zero to $\mathcal D$. We define $$ \mathbb G := \mathbb R \setminus \mathcal D. $$ From a generative perspective, ghost numbers are further classified along two independent axes. According to the digit law, one distinguishes *pure* ghost numbers (generated by a uniform i.i.d. process), *spurious* ghost numbers (generated by a non-uniform but still full-alphabet i.i.d. process), and *degenerate* ghost numbers (generated by an i.i.d. process supported on a proper subset of the alphabet). Independently, one distinguishes *natural* ghost numbers (arising directly from an i.i.d. process) from *constructed* ghost numbers (obtained by finite algorithmic transformations of already given objects). The paper establishes the following results: 1. $\mathcal D$ is a subfield of $\mathbb R$, whereas $\mathbb G$ is not closed under the four arithmetic operations. 2. Every $g\in\mathbb G$ is transcendental over $\mathcal D$, in the sense that it is not a root of any non-zero polynomial with coefficients in $\mathcal D$. 3. The natural generative process for ghost numbers is given by the infinite digit sequence; for definable numbers, such a representation is only one among many. 4. The zig-zag constants are algebraic for every finite base; in the continuous limit a transcendental constant appears, whereas for $b=2$ the natural process yields the golden ratio. 5. Antipodal ghost constellations exhibit maximal individual incompressibility under exact collective constraints. 6. The digit process realises discrete white noise exactly in base $b$. 7. The theory of constellations depends on the collective permutation of digits across the channels of the constellation, rather than on the physical device used to implement it. 8. Two independent realisations of the same law may coincide for an arbitrarily large but finite number of steps, but the probability that they coincide at every step is zero. 9. A concrete realisation of a constellation cannot be regenerated by any finite parameter-free description in the fixed language. The notion of a ghost number is metamathematical: it concerns parameter-free non-definability in the fixed background framework. It does not, in general, coincide with non-computability or with algorithmic randomness in the sense of Martin-Löf, although the i.i.d. process considered here almost surely produces realisations with prefixes of asymptotically maximal algorithmic complexity. The paper introduces no new axioms and does not modify any classical theorem. --- **Note on this version (April 2026)** In the original formulation of Section 8, the set $\mathbb G_{\mathrm{nat}}$ was defined as the set of all realisations of the uniform i.i.d. digit process, while the strong law of large numbers was implicitly invoked as if it held for every such realisation. This is incorrect: the strong law guarantees convergence only almost surely, i.e., on a subset of full measure, not pointwise for all elements of the sample space. To remedy this, the revised version distinguishes between $\mathbb G_{\mathrm{iid}}$ (all realisations) and $\mathbb G_{\mathrm{typ}}$ (the typical realisations satisfying the frequency condition). The construction of $\mathbb G_{\mathrm{fin}}$ and $\mathbb G_{\mathrm{anti}}$ is then carried out using $\mathbb G_{\mathrm{typ}}$. The mathematical content of the anti-generative construction — a dense, measure-zero class of ghost numbers structurally unreachable by the i.i.d. process — remains unchanged and is now rigorously justified. All other sections are unaffected. --- **Keywords**: ghost numbers, non-definable reals, transcendence over $\mathcal D$, constellations, algorithmic randomness, white noise, anti-generative numbers