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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: Sat, 9 Nov 2024 15:58:24 -0500
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On 11/9/24 2:50 PM, olcott wrote:
> On 11/9/2024 1:32 PM, Alan Mackenzie wrote:
>> olcott <polcott333@gmail.com> wrote:
>>
>>> The assumption that ~Provable(PA, g) does not mean ~True(PA, g)
>>> cannot correctly be the basis for any proof because it is only
>>> an assumption.
>>
>> It is an assumption which swifly leads to a contradiction, therefore must
>> be false. 
> 
> You just said that the current foundation of logic leads to a 
> contradiction. Too many negations you got confused.
> 
> When we assume that only provable from the axioms
> of PA derives True(PA, g) then (PA ⊢ g) merely means
> ~True(PA, g) THIS DOES NOT LEAD TO ANY CONTRADICTION.
> 
>> But you don't understand the concept of proof by
>> contradiction, and you lack the basic humility to accept what experts
>> say, so I don't expect this to sink in.
>>
> 
> 
>>>> We know, by Gödel's Theorem that incompleteness does exist.  So the
>>>> initial proposition cannot hold, or it is in an inconsistent system.
>>
>>> Only on the basis of the assumption that
>>> ~Provable(PA, g) does not mean ~True(PA, g)
>>
>> No, there is no such assumption.  There are definitions of provable and
>> of true, and Gödel proved that these cannot be identical.
>>
> 
> *He never proved that they cannot be identical*
> 
> The way that sound deductive inference is defined
> to work is that they must be identical.

Nope, becuase

TRUE is based on ANY sequence of steps, including an infinite sequence.

PROVABLE is based on only a FINITE sequence of steps.

> 
> A conclusion IS ONLY true when applying truth
> preserving operations to true premises.

Which might be infinite, and thus not a proof.

> 
> It is very stupid of you to say that Gödel refuted that.
> 

Because he did, for the actual definitions, not your false one.

Sorry God you are that can't undetstand what a infinite thing is.