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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: The philosophy of logic reformulates existing ideas on a new
 basis --- infallibly correct
Date: Mon, 11 Nov 2024 08:51:20 -0600
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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.
What ceases to be possible is proving that a false claim is true.

Within the premise that "elephants are dead mice" it can be
proved that {elephants are dead mice} the non-truth of that
expression is preserved.

    Let {T} be such a theory. Then the elementary
    statements which belong to {T} we shall call the
    elementary theorems of {T}; we also say that
    these elementary statements are true for {T}.
    Thus, given {T}, an elementary theorem is an
    elementary statement which is true.
    https://www.liarparadox.org/Haskell_Curry_45.pdf

Haskell Curry is referring to a set of expressions that are
stipulated to be true in T.

We define True(L, x) to mean x is a necessary consequence of
the Haskell Curry elementary theorems of L.
(Haskell_Curry_Elementary_Theorems(L) □ x) ≡ True(L, x)

> 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.
> 

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer