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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory,sci.logic
Subject: Re: H(D,D) cannot even be asked about the behavior of D(D) --- Truth
 Itself is not Broken. (Just misunderstood)
Date: Sat, 15 Jun 2024 11:24:43 -0400
Organization: i2pn2 (i2pn.org)
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On 6/15/24 11:17 AM, olcott wrote:
> On 6/15/2024 8:52 AM, Richard Damon wrote:
>> On 6/15/24 8:26 AM, olcott wrote:
>>> On 6/15/2024 6:48 AM, joes wrote:
>>>> Am Fri, 14 Jun 2024 21:39:50 -0500 schrieb olcott:
>>>>
>>>>> The key aspect of all of this is that if the halting problem is 
>>>>> correct
>>>>> then truth itself is fundamentally broken. Since truth itself cannot
>>>>> possibly be fundamentally broken it must be fallible human 
>>>>> understanding
>>>>> of truth that is actually broken.
>>>> I've got bad news for you, and you're a century late. Gödel proved that
>>>> not all true statements are provable. It sure would have been nice.
>>>>
>>>
>>> He didn't even prove this.
>>> He proved that a statement that can be expressed in PA
>>> cannot be proving in PA that is not true in PA yet can be
>>> proved in matamath thus is true in metamath.
>>
>> No, you are just showing you don't know what you are talking about.
>>
>> You have admitted that you don't even understand the actual statement 
>> he was using, but can only understand it as the simplication through 
>> implication in the meta-thoery.
>>
>> G, the statement about the non-existance of a Natural Number 'g' that 
>> satisfied the specified relationship.
>>
>> It can be shown (in the meta-theory) that no such number can exist, so 
>> G must be true. Either the number 'g' exists or it doesn't so their 
>> can be no middle ground, and it if is shown (in the meta theory) that 
>> if such a number 'g' existed, then we could build a proof (as encoded 
>> in the finite number 'g') in PA that PROVES CONCLUSIVELY no such 
>> number exist. 
> 
>> There can not be a number that proves that itself doesn't exist, so 
>> there must not be such a number.
>>
> 
> Isomorphically there can be no proof that proves itself doesn't exist, 
> so there must not be such a proof.
> 

So?

In the meta-math there IS a proof that there is no number 'g' that 
satisfies that relationship, but that isn't doing a proof about ITSELF.

The key is that if we find a number 'g' in PA that satisfies the 
relationship, then via the operations in the meta-math, we can CONSTRUCT 
an actual proof in PA that such a number can not exist.

This becomes the proof, in the meta-math, that no number 'g' can exist.

In PA alone, we can still establish that fact, but only with infinite 
work, so we can't form a proof of that fact.