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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory,sci.logic,comp.ai.philosophy
Subject: Re: Annotated Breakdown: "computing the mapping from an input"
Date: Sun, 20 Apr 2025 17:17:43 -0500
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On 4/20/2025 3:57 PM, Mr Flibble wrote:
> 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. This blurs the boundary between object-level and meta-level
> semantics.
> 3. Invoke D on Itself
> Evaluate D(D), which leads to the contradiction:
>      - If H(D, D) says D halts → D(D) loops
>      - If H(D, D) says D loops → D(D) halts
> 
> Category-Theoretic View: Here the category error is fully exposed. The
> object D is passed into H and simultaneously defined in terms of H’s
> result on itself. This self-referential construct crosses semantic layers:
> a program is both subject and evaluator. In type-theoretic terms, this is
> akin to an ill-formed expression — a form of circular logic not grounded
> in a well-defined semantic domain.

Yes we perfectly agree on this, when the input
can possibly do the opposite of whatever value
that its decider returns the question posed to
H is self-contradictory: H(D) Does your input halt?

D cannot actually do the opposite of whatever H returns
when H is a simulating termination analyzer. Instead D
keeps calling H in recursive simulation until H aborts
its simulation of D. Thus mapping THE ACTUAL INPUT STRING
(not any damn thing else) to the behavior that it species
(including D simulating itself simulating D) conclusively
proves that H is correct to reject D as non-halting.

Woefully ignorant people that have no idea what the Hell
"computing the mapping from an input" is, how it works, or
why it is required may disagree.

> 4. Conclude H Cannot Exist
> The contradiction implies that no such universal halting decider H can
> exist.
> 
> Category-Theoretic View: From this perspective, the contradiction arises
> not from an inherent limitation of deciders per se, but from allowing
> semantically invalid constructs. D(D) is seen as undefined or outside the
> valid domain of discourse — not a legitimate program-input pair, but a
> malformed self-referential statement.

Yes, good job you have quickly gained deep insight.

> 5. Interpretation
> The standard interpretation is that the Halting Problem is undecidable —
> there is no algorithm that can determine for all programs and inputs
> whether the program halts.
> 
> Category-Theoretic View: The undecidability arises only when the system
> permits semantically malformed constructions. If the language of

Yes and all undecidability seems to be from failing to reject
erroneous input.

> computation is refined to exclude category errors — such as programs that
> attempt to reference or reason about their own evaluation in this way —
> then within that well-formed subset, halting may be decidable or at least
> non-contradictory.

This same thing equally applies to the Tarski Undefinability Theorem.
https://liarparadox.org/Tarski_275_276.

Truth is a necessary consequence to applying the truth
preserving operation of semantic entailment to the set
of basic facts (cannot be derived from other facts)
expressed in language.

All of undecidability comes from breaking the above rules.

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer