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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: The philosophy of logic reformulates existing ideas on a new basis --- infallibly correct
Date: Tue, 12 Nov 2024 12:44:08 +0200
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On 2024-11-11 14:51:20 +0000, olcott said:

> On 11/11/2024 4:33 AM, Mikko wrote:
>> On 2024-11-11 04:41:24 +0000, olcott said:
>> 
>>> On 11/10/2024 10:03 PM, Richard Damon wrote:
>>>> On 11/10/24 10:08 PM, olcott wrote:
>>>>> On 11/10/2024 3:52 AM, Mikko wrote:
>>>>>> On 2024-11-09 18:05:38 +0000, olcott said:
>>>>>> 
>>>>>>> On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
>>>>>>>> olcott <polcott333@gmail.com> wrote:
>>>>>>>>> On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
>>>>>>>>>> olcott <polcott333@gmail.com> wrote:
>>>>>>>>>>> On 11/9/2024 5:01 AM, joes wrote:
>>>>>>>>>>>>>>>>>>>> On 11/8/24 12:25 PM, olcott wrote:
>>>>>>>> 
>>>>>>>>>>>>>>>>>>>>> That formal systems that only apply truth preserving operations
>>>>>>>>>>>>>>>>>>>>> to expressions of their formal language that have been
>>>>>>>>>>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is proven
>>>>>>>>>>>>>>>>>>>>> to be over-your-head on the basis that you have no actual
>>>>>>>>>>>>>>>>>>>>> reasoning as a rebuttal.
>>>>>>>>>>>> Gödel showed otherwise.
>>>>>>>> 
>>>>>>>>>>> That is counter-factual within my precise specification.
>>>>>>>> 
>>>>>>>>>> That's untrue - you don't have a precise specification.  And even if you
>>>>>>>>>> did, Gödel's theorem would still hold.
>>>>>>>> 
>>>>>>>>>>> When truth is only derived by starting with
>>>>>>>>>>> truth and applying truth preserving operations
>>>>>>>>>>> then unprovable in PA becomes untrue in PA.
>>>>>>>> 
>>>>>>>>>> No.  Unprovable will remain.
>>>>>>>> 
>>>>>>>>> *Like I said you don't pay f-cking attention*
>>>>>>>> 
>>>>>>>> Stop swearing.  I don't pay much attention to your provably false
>>>>>>>> utterances, no.  Life is too short.
>>>>>>>> 
>>>>>>> 
>>>>>>> That you denigrate what I say without paying attention to what
>>>>>>> I say <is> the definition of reckless disregard for the truth
>>>>>>> that loses defamation cases.
>>>>>>> 
>>>>>>>> Hint: Gödel's theorem applies in any sufficiently powerful logical
>>>>>>>> system, and the bar for "sufficiently powerful" is not high.
>>>>>>>> 
>>>>>>> 
>>>>>>> Unless it is stipulated at the foundation of the notion of
>>>>>>> formal systems that ~Provable(PA, g) simply means ~True(PA, g).
>>>>>>> 
>>>>>>>>> Unprovable(L,x) means Untrue(L,x)
>>>>>>>>> Unprovable(L,~x) means Unfalse(L,x)
>>>>>>>>> ~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
>>>>>>>> 
>>>>>>>> If you're going to change the standard meaning of standard words, you'll
>>>>>>>> find communicating with other people somewhat strained and difficult.
>>>>>>>> 
>>>>>>> 
>>>>>>> ZFC did the same thing and that was the ONLY way
>>>>>>> that Russell's Paradox was resolved.
>>>>>>> 
>>>>>>> When ~Provable(PA,g) means ~True(PA,g) then
>>>>>>> incompleteness cannot exist.
>>>>>> 
>>>>>> But it doesn't. "Provable(PA,g)" means that there is a proof on g in PA
>>>>>> and "~Provable(PA,g)" means that there is not. These meanings are don't
>>>>>> involve your "True" in any way. You may define "True" as a synonym to
>>>>>> "Provable" but formal synonyms are not useful.
>>>>>> 
>>>>> 
>>>>> 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.
>>>> 
>>> 
>>> Proof(Olcott) means a sequence of truth preserving operations
>>> that many not be finite.
>> 
>> With a hyperfinite sequnce it is possible to prove a false claim.
>> 
> 
> It will always be possible to merely prove a false claim.

Only if it is proven in an unsound system.

> What ceases to be possible is proving that a false claim is true.

Even that can be provable in an unsound system. At least it is
provable in an incosstent system that can express the claim.

And of course allowing hyperfinite proofs can break an otherwise
sound system:

>> The most obvious truth preserving operation is the identity operation.
>> Its result is the same as its premise, so the truth valure of the
>> result must be the same as the truth value of the premise. So we
>> can form a hyperfinite sequence
>> 
>> 1 = 1, 1 = 1, 1 = 1, ... , 1 = 2, 1 = 2, 1 = 2
>> 
>> where ... denotes infinitely manu intermedate steps. The first equation
>> is true, every other equation is as ture as the one before it and the
>> last equation is false.


-- 
Mikko