Re: H(D,D) cannot even be asked about the behavior of D(D) --- Truth Itself is not Broken. (Just misunderstood)

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.