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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: The philosophy of logic reformulates existing ideas on a new
 basis --- infallibly correct
Date: Mon, 11 Nov 2024 10:01:02 -0500
Organization: i2pn2 (i2pn.org)
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On 11/11/24 8:55 AM, olcott wrote:
> On 11/10/2024 10:03 PM, Richard Damon wrote:
>> On 11/10/24 10:07 PM, olcott wrote:
>>> On 11/10/2024 4:19 PM, Alan Mackenzie wrote:
>>>> olcott <polcott333@gmail.com> wrote:
>>>>> On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
>>>>>> olcott <polcott333@gmail.com> wrote:
>>>>>>> On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
>>>>
>>>>>> [ .... ]
>>>>
>>>>>>>> I have addressed your point perfectly well.  Gödel's theorem is 
>>>>>>>> correct,
>>>>>>>> therefore you are wrong.  What part of that don't you understand?
>>>>
>>>>>>> YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
>>>>>>> NOT GET RID OF INCOMPLETENESS.
>>>>
>>>>>> The details are unimportant.  Gödel's theorem is correct.
>>>>
>>>>> In other words you simply don't understand these
>>>>> things well enough ....
>>>>
>>>> Not at all.  It's you that doesn't understand them well enough to 
>>>> make it
>>>> worthwhile trying to discuss things with you.
>>>>
>>>>> .... to understand that when we change their basis the conclusion
>>>>> changes.
>>>>
>>>> You're at too high a level of abstraction.  When your new basis has
>>>> counting numbers, it's either inconsistent, or Gödel's theorem 
>>>> applies to
>>>> it.
>>>>
>>>
>>> Finally we are getting somewhere.
>>> You know what levels of abstraction are.
>>>
>>>>> You are a learned-by-rote guy that accepts what you
>>>>> memorized as infallible gospel.
>>>>
>>>> You're an uneducated boor.  So uneducated that you don't grasp that
>>>> learning by rote simply doesn't cut it at a university.
>>>>
>>>>>> Your ideas contradict that theorem.
>>>>
>>>>> When we start with a different foundation then incompleteness
>>>>> ceases to exist just like the different foundation of ZFC
>>>>> eliminates Russell's Paradox.
>>>>
>>>> No.  You'd like it to, but it doesn't work that way.
>>>>
>>>> [ .... ]
>>>>
>>>>>> Therefore your ideas are incorrect.  Again, the precise details are
>>>>>> unimportant,
>>>>
>>>>> So you have no clue how ZFC eliminated Russell's Paradox.
>>>>> The details are unimportant and you never heard of ZFC
>>>>> or Russell's Paradox anyway.
>>>>
>>>> Russell's paradox is a different thing from Gödel's theorem.  The 
>>>> latter
>>>> put to rest for ever the vainglorious falsehood that we could prove
>>>> everything that was true.
>>>>
>>>
>>> Ah so you don't understand HOW ZFC eliminated Russell's Paradox.
>>>
>>> We can ALWAYS prove that any expression of language is true or not
>>> on the basis of other expressions of language when we have a coherent
>>> definition of True(L,x).
>>
>> No, we can't.
>>
>> We can sometimes prove it is true if we can find the sequence of steps 
>> that establish it.
>>
>> We can sometime prove it is false if we can find the sequence of steps 
>> that refute it.
>>
>> Since there are potentially an INFINITE number of possible proofs for 
>> either of these until we find one of them, we don't know if the 
>> statement IS provable or refutable.
>>
>> Your problem is you think that knowledge and truth are the same, but 
>> knowledge is only a subset of truth, and there are unknown truths, and 
>> even unknowable truths in any reasonably complicated system.
>>
>> Part of your issue is you seem to only think in very simple systems 
>> where exhaustive searching might actually be viable.
>>
>>>
>>> That Gödel relies on True(meta-math, g) to mean True(PA, g)
>>> is a stupid mistake that enables Incomplete(PA) to exist.
>>>
>>>
>>
>> Which just shows you don't understand how formal systems, and their 
>> meta-systems are constructed.
>>
> 
> It does not matter how they are constructed the only
> thing that matters is the functional end result.

OF course it does. If you don't understand the rules by which a system 
was constructed, you can't know what you can do in the system.

Yes, an ordinary user of a system may not need to know the gritty 
details of the system, but to claim it is not logical, requires going 
into the rules to find the error, otherwise the error is more apt to be 
in the "logic" that the user is trying to apply. (Like what WM has been 
doing).

> 
> *When we construe True(L,x) this way*
> When g is a necessary consequence of the Haskell Curry
> elementary theorems of PA (Thus stipulated to be true in PA)
> then and only then is g is True in PA.

But G *IS* a necessary consequence of the axioms of PA. Yes, it needs an 
infinite number of steps, but it is demonstrable by them.

That 0 does not satisfy the PRR, is a simple matter of the mathematics 
created by those axioms of PA.

That 1 does not satisfy the PRR, is a simple matter of the mathematics 
created by those axioms of PA.

That 2 does not satisfy the PRR, is a simple matter of the mathematics 
created by those axioms of PA.

That any given natural number g does not satisfy the PRR, is a simple 
matter of the mathematics created by those axioms of PA, but we have to 
evaluate this individually for each number g.

Thus, we have a chain of necessary consequences, infinite in length, 
that shows that the statement G is true, G being that there is no number 
g that satisfies that particular PRR.


The fact that we can prove, in MM, the fact that in general, no g can 
satisfy the PRR in a finite number of steps, doesn't negate that it is a 
necessary consequence in PA, it just takes longer there.

In MM, we can also prove that there is no finite sequence of steps in PA 
that would show it to be a necessary consequence, and the PRR was 
constructed in MM (using the operation available in PA) such that any 
finite proof in PA of that statement could be encoded into a number 
(which exist in PA) that would satisfy that PRR which can be processed 
in PA. Since the existance of such a number both proves that a number 
satisfing the PRR exists, and that no such number can exist, there can't 
be such a number.

> 
> https://www.liarparadox.org/Haskell_Curry_45.pdf
> (Haskell_Curry_Elementary_Theorems(PA) □ g) ≡ True(PA, g)
> 
> If there is no sequence of truth preserving operations
> in PA from its Haskell_Curry_Elementary_Theorems to g
> then it can be construed that g is simply not true in PA.
> Incorrect(PA,g) ≡ (True(PA, g) ∧ True(PA, ~g))

But I just showed it, AGAIN to you, so, you claim was refuted before you 
said it, so is just a lie.
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