On 5/11/25 9:21 AM, Mr Flibble wrote:
Flibble’s Leap: Why Behavioral Divergence Implies a Type Distinction in
the Halting Problem
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Summary
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Flibble argues that the Halting Problem's undecidability proof is built on
a category (type) error: it assumes a program and its own representation
(as a finite string) are interchangeable. This assumption fails under
simulating deciders, revealing a type distinction through behavioral
divergence. As such, all deciders must respect this boundary, and
diagonalization becomes ill-formed. This reframing dissolves the paradox
by making the Halting Problem itself an ill-posed question.
1. Operational Evidence of Type Distinction
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- When a program (e.g., `DD()`) is passed to a simulating halt decider
(`HHH`), it leads to infinite recursion.
But it doesn't a a "decider", must, BY DEFINITION, always return in finite time, so a decider can not let itself get stuck in "infinite recursion".
- This behavior differs from direct execution (e.g., a crash due to a
stack overflow).
No, because if the HHH that DD calls *IS* a decider, it will not get into the infinite recursion.
- This divergence shows that the program (as code) and its representation
(as data) are operationally distinct.
Nope, it shows that you concept of a decider is incorrect.
What is a caterory error is the assumption that a given program can be both a correct simulator and a decider.
Yes the program and its representation are distinct entities, but entities in a well defined relationship, allowing the translation of requirements across it.
I guess you think computers can't process Numbers, Text, Sound or Images, as none of these are things that the processor actually directly processes.
- Therefore, treating them as the same "type" (as Turing's proof does)
leads to inconsistency.
Nope, the error is the assumption that a program can be both a correct decider and a correct simulator at the same time.
2. Generalization to All Deciders
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- If simulation reveals this type mismatch, then no valid decider can rely
on conflating program and representation.
But there isn't a type mismatch
- Diagonalization (e.g., defining D(x) = if H(x,x) then loop else halt)
necessarily crosses the type boundary.
What "Type Boundry". You haven't defined a "type".
- The contradiction in the Halting Problem arises from this type error,
not from inherent undecidability.
But you haven't defined what type was violated, only that you have found it impossible to do what you want to do.
- Hence, the Halting Problem is ill-defined for self-referential input.
Nope, it shows that the concept of a simulating halt decider, which requires the machine to be both a correct simulator and a correct halt decider is the ill-defined term based on a self-contradictory definition.
3. Comparisons to Other Formal Systems
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- In type-theoretic systems (like Coq or Agda), such self-reference is
disallowed:
- Programs must be well-typed.
- Self-referential constructs like `H(x, x)` are unrepresentable if they
would lead to paradox.
- In category theory, morphisms must respect domain/codomain boundaries:
- Reflexive constructions require stratification to avoid logical
inconsistency.
Then those system are just proved to be not Turing Complete, and thus not applicable to Computation Theory.
If you can not take a program D, and from that compute a representation of that <D>, such that you can make a simulator that can process that <D> to recreate the behavior of D, and then give that input to D itself, then you compulation system is just inferior to the Turing Complete system.
4. Conclusion
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- The Halting Problem depends on self-reference and diagonalization.
Nope, the problem itself just asks if a finite algorithm expressed in a Turing Complete logic system can take as its input a representation of any such finite algorithm and a data input for it, and determine if that finite algorithm / data combination would complete in finite time when performed.
This is a valid question, as for *ANY* possible input, there is a correct answer, and every program / input combination, its behavior is FULLY determined and *WILL* either halt in a finite number of steps, or continue to run for an unbounded number of steps.
- If these constructs are blocked due to type/category errors, the proof
breaks down.
No, it says your type system is in error, as the problem has no type or category errors.
- Thus, the Halting Problem isn’t undecidable—it’s malformed when it
crosses type boundaries.
But it doesn't cross the type boundries of the system it is in.
- This is not a refutation of Turing, but a reformulation of the problem
under more disciplined typing rules.
Which just shows your typing rules are not compatible with Turing Complete computation systems.
This model mirrors how Russell’s Paradox was avoided in modern logic: not
by solving the contradiction, but by redefining the terms that made the
contradiction possible.
If you want to try to do that, you need to do like Zermelo did, and work out a fully defined alternate theory that was still useful for the problems desired to be done.
For most people, the fact that some problems are undecidable isn't a big problem, as the simple counting argument shows that it is expected. To avoid that problem, you are going to need to drastically reduce the space of what a computation can do.
That is what ZFC did. The problem with the older Set Theory is that it allowed virtually unrestricted rules to create a set, and this is what lead to the contradictions. By defining a different set of rules for how to build a set, and showing that it could still handle the vast majority of the real problems that people wanted to use, made it a viable set theory.
IF you want to try to define rules about what can be done, go ahead and try. Remember, it needs to be a FORMAL definition, which fully defines what can be done. It also needs to result in a system powerful enough to be useful for a lot of the current cases.
Go ahead and try to define it.
Re: Flibble’s Leap: Why Behavioral Divergence Implies a Type Distinction in the Halting Problem