Re: Refutation of Turing’s 1936 Halting Problem Proof Based on Self-Referential Conflation as a Category (Type) Error

On 2025-04-25 10:31, olcott wrote:

Once we understand that Turing computable functions are only
allowed to derived their outputs by applying finite string
operations to their inputs then my claim about the behavior
of DD that HHH must report on is completely proven.

You're very confused here.
Computable functions are *functions*. That is, they are mappings from a domain to a codomain, neither of which are required to be strings. Functions don't involve finite string operations at all.
What distinguishes a computable function from other functions is that it is possible to implement an algorithm which computes that function. That algorithm might involve string operations, but the algorithm is not the function anymore than the function is the algorithm.
The halting *function* is a mapping from the set of computations (not strings) to the set of boolean values (also not strings). A given computation maps to true if and only if it is finite.
A halt *decider* (were such a thing possible) would be an algorithm which computes the halting function. Such an algorithm (assuming we're talking in terms of Turing Machines) would have to encode the elements of the halting function's domain as strings, but the mapping it computes would still be from the computations which those strings represent to their associated boolean values.
André
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