Re: Nature of undecidable halting ---Handling undecidable inputs

On 5/17/24 12:49 PM, olcott wrote:

On 5/17/2024 5:04 AM, Mikko wrote:

On 2024-05-16 13:20:48 +0000, joes said:
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Am Thu, 16 May 2024 13:42:41 +0300 schrieb Mikko:

On 2024-05-15 15:06:26 +0000, olcott said:

I refer to transitioning through a specific state to indicate a
specific halt status value, for Turing Machines.

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That does not satisfy the usual definition of "halt decider". However,
we could accept that as a solution to the halting problem if one could
prove that there is a Turing machine that can indicate halting or
non-halting that way for all computations.
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However, it is possible to prove that every Turing machine that
indicates halting that way fails to indicate correctly at least some
computations.

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Are these all of the liar paradox kind, such that one could easily
exclude them? Or do they form a more interesting class?

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The discussions hera are mainly about the liar paradox or Quine paradox
kind. They ara not always easy to exclude, and one can always modify the
code so that exclusion becomes harder but behaviour remanis the same.
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 Expressions that are {true on the basis of meaning} are ONLY
(a) A set of finite string semantic meanings that form an accurate model
     of the general knowledge of the actual world.
 (b) Expressions derived by applying truth preserving operations to (a).
 The above algorithm specifies True(L,x) and False(L,x) defined
as True(L, ~x).
 The above expressions include all of expressions of math, logic and
computations specified as finite strings. The above True(L,x) combined
with False(L,x) seems to screen out any any all undecidable inputs.
Truthbearer(L,x) ≡ (True(L,x) ∨ False(L,x)) else type mismatch error.
 

And you don't seem to be able to work forward to see that if for p defined in L as ~True(L, p) is made false by that logic, that makes p defined as the equivalent of ~false, and thus p is defined as the equivalent of true, and thus you have made True(L, true) to be false, and you system it shown to be inconsistant.

 

There are also very different problems that are known to be uncomputable,
e.g., the problem whether a sentence in the language of the first order
goup theory is a theorem of that theory -- nothing like the liars paradox
is possible in that language.
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