Re: Annotated Breakdown: Linz's Halting Problem Proof and the Category Error Perspective

On 2025-04-20 20:57:30 +0000, Mr Flibble said:

This is a step-by-step outline of Linz's classical proof of the
undecidability of the Halting Problem, with commentary from the
perspective of a category-theoretic critique. This perspective suggests
that certain steps in the proof involve category errors, where roles and
types of entities are improperly mixed — for example, treating a program
and a meta-level decider as interchangeable.
1. Assume a Halting Decider Exists
Linz begins by assuming the existence of a function H(P, x) that can
determine whether program P halts on input x.
 Category-Theoretic View: This assumption does not yet involve any category
error. It describes a standard computational decider working over ordinary
program-input pairs.
2. Define a Contradictory Program D(P)
Construct a new program D such that:
    if H(P, P) reports 'halts', then D(P) loops forever;
    if H(P, P) reports 'loops', then D(P) halts.
 Category-Theoretic View: This step begins to introduce potential category
confusion. The function D is now being constructed specifically to act in
contradiction to H's analysis of P on itself, blending the role of program
and predicate.

There is no category error in blending the role of program and predicate.
Ther difference between program and predicate is not in their roles but
in their nature. A program specifies a method. A predicate does not specify
any method.
--
Mikko