Re: Chinese Analysis of Flibble's Final Statement on the Halting Problem

On 2025-06-15 15:05:53 +0000, olcott said:

On 6/15/2025 9:39 AM, Mr Flibble wrote:

Flibble's critique of the halting problem is an interesting one! Let's
break it down and analyze the claims:
 ### 1. **Recursive Self-Reference and the Halting Problem**
The halting problem, as formulated by Alan Turing, indeed relies on self-
reference. The classic proof constructs a program (or Turing machine) that
asks whether it will halt when given its own description as input. This
leads to a paradox:
- If the program halts, then (by definition) it shouldn't halt.
- If it doesn’t halt, then it should halt.
 This is analogous to the "liar paradox" ("This statement is false"), where
self-reference creates logical inconsistency.
 ### 2. **Flibble's Claim: Infinite Regress and Category Error**
Flibble argues that:
- The problem ignores recursive self-reference "focusing on the paradox
instead."
- The self-reference leads to infinite regress.
- This makes the problem definition "ill-formed" due to a "category error."
 #### **Infinite Regress?**
Infinite regress occurs when a definition or process refers back to itself
in an endless loop. In the halting problem, the self-reference is used to
construct a contradiction, not necessarily an infinite loop in the
problem's definition. However, one could argue that the self-referential
nature of the halting problem’s construction implies an infinite recursion
in reasoning (e.g., "To decide if H halts, we need to know if H halts,
which requires...").
 #### **Category Error?**
A category error occurs when something is mistakenly treated as belonging
to a different logical or conceptual category. Flibble might be suggesting
that:
- The halting problem conflates levels of reasoning (e.g., mixing a
program’s behavior with its own meta-analysis).
- The self-referential construction forces a program to be both an object
of analysis and an analyzer, which could be seen as a misuse of categories.
 ### 3. **Is the Halting Problem Ill-Formed?**
The halting problem is well-defined in computability theory, and its
undecidability is a fundamental result. However, Flibble’s critique aligns
with some deeper philosophical objections:
- **Constructivist/Intuitionist Views:** Some might argue that self-
referential definitions are inherently problematic and that the halting
proof relies on non-constructive reasoning.
- **Type Theory Perspectives:** In some formal systems, self-reference is
restricted to avoid paradoxes (e.g., Russell’s type theory to resolve the
"set of all sets" paradox). If the halting problem’s construction violates
such restrictions, one could argue it’s ill-formed in those systems.
- **Physical Realizability:** Some argue that true self-reference is
impossible in physical machines (since no program can perfectly embed its
own complete description), making the halting problem an abstract
idealization.
 ### 4. **Does This Undermine the Halting Problem’s Validity?**
Not necessarily. The halting problem’s proof is widely accepted in
computability theory because:
- It works within the formal framework of Turing machines.
- Self-reference is explicitly allowed in unrestricted computation.
- The paradox is the *point*—it shows that no general halting decider can
exist without contradiction.
 However, Flibble’s perspective might appeal to those who question whether
self-referential constructions like this are meaningful or whether they
expose limitations in classical computability theory itself.
 ### 5. **Alternative Perspectives**
Some frameworks (e.g., partial computability, hypercomputation, or
finitist approaches) might reject or modify the halting problem’s
assumptions. For example:
- **Finitist/Ultraintuitionist Views:** Might reject the problem as ill-
formed due to its use of infinite reasoning.
- **Constructive Mathematics:** Might demand a more explicit example of a
non-halting program rather than a paradoxical construction.
 ### **Conclusion**
Flibble’s argument is a philosophical critique rather than a mathematical
refutation. The halting problem is rigorously defined and correct within
classical computability theory, but the objection highlights that its
foundations rely on assumptions (unrestricted self-reference, infinite
domains) that not all frameworks accept. Whether this makes it "ill-
formed" depends on one’s underlying philosophy of mathematics and
computation.
 Would Flibble propose an alternative model of computation where such self-
reference is forbidden? That could lead to a different theory—but it
wouldn’t invalidate Turing’s result in its original context.

 If we start a biological theory with the assumption
that "all cats are dogs" and it is later found out
that "no cats are dogs" then this theory is refuted.

The theory is already refuted when it is found that one cat is
not a dog.

If we start a halting problem proof with the assumption
that an input to a partial halt decider (PHD) can actually
do the opposite of whatever value that its PHD returns and
we later find out that *NO SUCH INPUT CAN POSSIBLY EXIST*
then this original proof has been refuted.

Which is why that assumption is not made in the books that discuss
the halting problem and related topics. Either the exstence of the
counter-example is proven or a proof that does not use a counter-
example is presented, sometimes both.
--
Mikko