On 5/25/25 5:33 AM, Mr Flibble wrote:
Can Moving the Paradox Code into the SHD Layer Reintroduce the Halting Problem? ================================================================================
Question:
---------
In the neos solution (Flibble's typed SHD model), programs are split into
two layers:
- The SHD layer (meta-level) that analyzes programs.
- The program layer that cannot reference the SHD.
Why do "Programs" has a SHD layer, since most programs don't use a SHD,
and how can you tell if code *IS* a SHD. Why can't you just include a
copy of the code of the SHD in the program layer?
What happens if you move the paradoxical construction (e.g., D() = if H(D) then loop) into the SHD layer itself?
Analysis:
---------
1. Stratification Prevents Self-Reference (If Enforced) --------------------------------------------------------
- Typed SHD systems (like Coq, Agda, or neos with layered semantics) can prevent self-reference by using type stratification.
- In this setup, the SHD layer cannot analyze itself.
- The construction of `D'` within the SHD layer, which calls `H(D')`,
would be semantically ill-typed and rejected.
2. If SHDs Can Analyze Themselves, the Paradox Returns -------------------------------------------------------
- If SHDs are allowed to analyze other SHDs or themselves, you can reintroduce the diagonal construction:
D': if H(D') then loop forever; else halt;
- This results in the same contradiction as in the classical halting
problem proof.
Yes, which means if you can define the layers, you can't prove the
correctness of the provers, as nothing can prove your top level prover
to be correct.
Conclusion:
-----------
- Moving the paradox into the SHD layer only prevents the contradiction if the SHD layer is **itself** stratified and prevents self-analysis.
Yes, this is why Type Theory tend to define a hierarchy with an
arbitrary number of layers.
- The paradox is not avoided merely by code relocation—it is avoided by **semantic restriction**.
- The issue is not *where* the paradox is written, but *what is allowed*
in the system.
Final Verdict:
--------------
You can't escape the Halting Problem paradox simply by shifting the code
to a different layer. You must redefine your system's rules to prevent any form of self-reference in all layers, including the SHD meta-level.
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