A definability paradox about real numbers: if one attempts to enumerate all real numbers definable in a natural language, one can diagonalize this list to produce a new real number that is itself definable, producing a contradiction that exposes imprecision in informal definability.

Principle

Mixing informal, meta-linguistic descriptions of definability with the assumption that those descriptions form a countable list leads to an effective diagonal construction that yields a definable object not on the list, showing the ambiguity of 'definable' in natural language.

Demonstration

List English descriptions that purportedly define real numbers and enumerate them; construct a new real whose nth decimal differs from the nth number's nth decimal. The new real is described by this diagonal procedure, contradicting the claim that the original list was complete.

Misapplication

Treating natural-language descriptions as formally precise definitions or assuming that informal definability behaves like formal definability in a fixed formal system; or using the paradox to claim that formal definability is incoherent.

Consequence

Shows the need to formalize notions of definability and distinguish object language from meta-language; in formal settings (e.g., definability in arithmetic or set theory) the paradox dissolves once the language and encoding are fixed.

Reversal

If 'definable' is made precise inside a fixed formal system with a clear coding, there is no paradox: definability becomes a formal property subject to standard diagonalization results but not the informal contradiction Richard exhibited.

Boundary

Pertains to informal, natural-language accounts of definability and to naive enumerations of definitions. It does not imply contradiction within a rigorously formalized theory where definitions and encodings are explicit.

Semantic Tension

Tension between the apparent countability of natural-language descriptions and the vagueness of what counts as a description; the paradox exploits this tension between informal discourse and formal encoding.

Synthesis

Richard's paradox highlights that informal talk of 'all definable numbers' is unstable: precise formalization of definability and coding removes the paradox, but the informal notion cannot safely be treated as if it were a formal, countable list.