On 7/12/2021 10:27 PM, Richard Damon wrote:
> On 7/12/21 3:51 PM, olcott wrote:
>> On 7/12/2021 10:33 AM, David Brown wrote:
>>> On 12/07/2021 16:25, olcott wrote:
>>>> On 7/12/2021 8:19 AM, Malcolm McLean wrote:
>>>>> On Monday, 12 July 2021 at 14:13:15 UTC+1, Andy Walker wrote:
>>>>>> On 10/07/2021 19:06, olcott wrote:
>>>>>>> At most I have only proved that the conventional proofs of the
>>>>>>> undecidability of the halting problem that rely on the Strachey form,
>>>>>>> are incorrect. This seems to include all textbook proofs.
>>>>>> As you know, Linz gives a quite different proof; and there are
>>>>>> several other proofs, inc one via "Busy Beaver" [see below].
>>>>>>
>>>>>> [...]
>>>>>>> Goldbach's Conjecture is merely undecided and thus not undecidable.
>>>>>> "Thus" is a step too far. But note the reference to Goldbach
>>>>>> in the Wiki article on "Busy Beaver".
>>>>>>
>>>>>> [...]
>>>>>>> Busy Beaver only seems intractable not undecidable.
>>>>>> It's undecidable whether an arbitrary TM is a BB; and the
>>>>>> BB function [ie, the "score" of a BB of given size] is uncomputable.
>>>>>> That's a stronger result than intractability. The proof that the
>>>>>> BB function is uncomputable is [reasonably] elementary, and does not
>>>>>> rely on the HP [or recursion]. If you had a "halting decider", then
>>>>>> you could find BBs by exhaustive [albeit intractable!] search and
>>>>>> thus compute the BB function; so there's yet another HP proof.
>>>>>>
>>>>> If you have a busy beaver function and a simulating decider, the busy
>>>>> beaver function tells you how many times you need to step the decider
>>>>> before you can be certain it will never halt. So halting is
>>>>> decideable if
>>>>> busy beaver is decideable.
>>>>>
>>>>>
>>>>
>>>> Flibble understands this:
>>>> What correct Boolean value can a TM return to an input that does the
>>>> opposite of whatever it decides? : neither, is excluded from the
>>>> solution set thus making the question itself incorrect.
>>>>
>>>
>>> You still don't understand this "proof by contradiction" concept, do you?
>>
>> We can say that the following expression is undecidable on the basis
>> that both true and false derive contradictions:
>> "This sentence is not true." That same sentence is used as the basis of
>> the Tarski undefinability theorem.
>>
>>
http://www.liarparadox.org/Tarski_247_248.pdf
>>
http://www.liarparadox.org/Tarski_275_276.pdf
>>
>>
>>
>> Although we have proven that it is undecidable on the basis of proof by
>> contradiction we are ignoring the key detail that the proof by
>> contradiction only succeeds because the expression of language is
>> self-contradictory, thus not a truth bearer and therefore erroneous.
>
> Does P(I) Halt IS a Truth Bearing Sentence, as P(I) will either Halt or
> it won't. We may not be able to prove that answer, but that is one
> fundamental property of the Higher level Logic that includes mathematics.
>
That is not the actual question, you leave out the key context that
makes the question incorrect. Ben played this same dishonest dodge since
2006. When you strip off the context of the question you change it into
an entirely different question.
Here is the actual question:
When we ask what Boolean value can a halt decider return to an input
that changes its behavior to contradict this value we cannot answer this
question because it is an incorrect type mismatch error question.
The answer is restricted to {true, false} thus excluding the correct
answer of “neither” making the question itself incorrect.
If we ask a man that has never been married:
Have you stopped beating you wife?
This is an incorrect question.
Every polar question lacking a correct yes/no answer is incorrect.
Every TM/input to a decision problem lacking a correct Boolean return
value is an incorrect TM/input for this decision problem.
The TM / input pairs that “prove” the halting problem is undecidable
have the same pathological self-reference(Olcott 2004) error as the
self-contradictory liar paradox.
It has been at least 2000 years since the liar paradox was discovered
and academicians still do not understand the because self-contradictory
sentences do not map to a Boolean value they are not truth bearers.
> If you want to keep to the concept that something is only true if you
> can prove it, then you must restrict yourself to only First Order Logic,
> or you WILL eventually run into an inconsistency in your logic system.
>
Not at all. Not in the least.
When Gödel's 1931 incompleteness theorem is translated into HOL that has
its own provability predicate thus no need for the purely extraneous
complexity of Gödel numbers then it is very obvious that it is modeled
after the liar paradox.
Gödel himself said that the liar paradox could be used as his
incompleteness basis.
there is also a close relationship with the “liar”
antinomy,14
We are therefore confronted with a proposition which
asserts its own unprovability
14 Every epistemological antinomy can likewise be
used for a similar undecidability proof.
This is HOL with its in provability predicate:
https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
I have examples that show the propositions that assert their own
unprovability are only undecidable because there is an infinite cycle in
their directed graph.
The problem is that like the liar paradox that have a vacuous truth
object. This sentence is not true. This sentence is not provable. Have
no truth object that they are true or provable about.
"It is true that this sentence is a dump truck."
has a truth object thus enabling the sentence to be evaluated as false.
"This sentence is true"
"This sentence is false"
"This sentence is provable"
Have no truth object, thus specify an infinite cycle in the directed
graph of their resolution.
> Yes, if you make that restriction to your logic, you can use that
> property, but you can not express the fullness of mathematics in your logic.
>
> This is well established.
>
I reestablished it correctly this time.