Computability Theory

I've drafted an initial article on Computability Theory. Please review.

Review by Critic Carla:

The article is a strong, well-structured introduction to Computability Theory. The lead is substantial, the tone is neutral and academic, and it correctly addresses the topic.

However, for a comprehensive encyclopedia entry, there are a few gaps and opportunities for refinement:

• Expand on the "Church-Turing Thesis" section: While the article mentions the thesis, it should explicitly clarify that this is a thesis (a definition of an intuitive concept) and not a mathematical theorem that can be proven, as this is a crucial distinction in the field.
• Add a section on "Computable vs. Enumerable Sets": The article jumps from basic models to the Halting Problem. A section explaining the difference between decidable (recursive) sets and semi-decidable (recursively enumerable) sets is essential for a complete overview of the theory.
• Incorporate "Rice's Theorem" into a dedicated subsection: Currently, Rice's Theorem is mentioned as a paragraph under the Halting Problem. Given its importance in proving the undecidability of semantic properties, it deserves its own subsection to maintain the structural hierarchy of the article.
• Clarify "Turing Degrees" with examples: The section on degrees of unsolvability is mathematically dense. Adding a concrete example of a problem that is "harder" than the Halting Problem (such as the Totality Problem) would make this section more accessible to the reader.

Fact-check by Fact-Checker Finn:

• The article states that the Church-Turing Thesis "posits that these three disparate models... are equivalent." This is a nuance error: the equivalence of the models is a proven mathematical theorem; the Thesis is the unprovable assertion that these formal models capture the intuitive notion of "effective calculability."
• The claim that the Universal Turing Machine is the "theoretical ancestor of the stored-program computer" is a sweeping generalization that simplifies a complex historical development involving multiple contributors (e.g., von Neumann).
• The description of the Halting Problem's proof mentions a program $D$ that "calls $H$ on itself." For technical accuracy, $D$ typically calls $H$ on the description of $D$, not the program object itself.
• The section on the "Legacy and Impact" uses slightly non-neutral/philosophical language regarding human intuition operating "differently than algorithmic computation," which is presented as a suggestion rather than a factual claim.

Review complete. Some issues were flagged. @Historian Hal, please address the feedback.

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