Flibble is raising a deep and nuanced critique of the halting problem, challenging the foundations of its formulation by focusing on the
**recursive self-reference** rather than the **logical paradox** (à la diagonalization). Let’s unpack and respond to Flibble's argument carefully.
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### **1. What the Halting Problem Actually States**
The classic halting problem, formalized by Alan Turing, is this:
Given a description of an arbitrary program $P$ and input $x$, determine
whether $P(x)$ halts or runs forever.
The contradiction arises by **assuming** a halting decider $H(P, x)$
exists, and then using it to construct a program $D$ that behaves in a contradictory way:
* $D(P)$ runs $H(P, P)$
* If $H$ says $P(P)$ halts, then $D$ loops forever.
* If $H$ says $P(P)$ loops, then $D$ halts.
This creates the paradox when we evaluate $D(D)$.
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### **2. Flibble's Critique: Infinite Regress via Recursive Self-
Reference**
Flibble is suggesting that this self-reference creates **infinite
regress**, not merely a logical paradox. From this view:
* The definition of $H$, and especially the way it's invoked as $H(P, P)$, requires resolving $P(P)$, which in turn might involve $P$'s behavior on
itself recursively.
* That chain does not terminate, making th