From Newsgroup: sci.logic

On 1/7/2026 6:05 AM, Mikko wrote:

On 06/01/2026 09:47, dart200 wrote:

-a ...

i think the actual problem is the TM computing is not sufficient to
describe all computable relationships.

It is not. Although we don't know any way compute what is not Turing computable,

In computability theory, a decider is a Turing machine
that halts for every input.[1] A decider is also called
a total Turing machine[2] as it represents a total function.

Because it always halts, such a machine is able to
decide whether a given string is a member of a formal
language. The class of languages that can be decided
by such machines is the set of recursive languages.

Given an arbitrary Turing machine, determining whether
it is a decider is an undecidable problem. This is a
variant of the halting problem, which asks for whether
a Turing machine halts on a specific input. https://en.wikipedia.org/wiki/Decider_(Turing_machine)

That seems a little silly because a TM that simply
halts on all inputs including the empty string is
said to have accepted that input. The "decision" in
this case is "don't make any decision, just accept".

*Here is a better definition by a*
https://yuvalfilmus.cs.technion.ac.il/
*PhD computer science Associate Professor*
Yuval Filmus

Intuitively, a decider should be a Turing machine that
given an input, halts and either accepts or rejects,
relaying its answer in one of many equivalent ways,
such as halting at an ACCEPT or REJECT state, or leaving
its answer on the output tape. https://cs.stackexchange.com/questions/84433/what-is-decider

Here is my own generalization across models
of computation.

All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.

Thus the simple way to determine what is not
computable is that any required result that
cannot be derived by applying finite string
transformation rules to specific inputs is
outside the scope of computation.

From this frame-of-reference "undecidability"
is not actual weakness or limit to computation.
It is merely forming a requirement that exceeds
the boundary of the scope of computation.

we can image a machine that can compute what a Turing
machine can't. Such machine can use a tape as an input and output
device as well as working storage just like Turing machine but has
additional instructions for operations that are not Turring computable.
While it is possible to imagine a halting decider for TUring machines implemented with an extended machine, a halting decider for all such
machines still requires that the decider can compute something that
none of the machines in its input domain can.

--
Copyright 2026 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>
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