Re: Mathematical incompleteness has always been a misconception --- Ultimate Foundation of Truth

On 3/2/25 5:01 PM, olcott wrote:

On 3/2/2025 3:25 PM, Richard Damon wrote:

On 3/2/25 4:16 PM, olcott wrote:

On 3/2/2025 2:11 PM, dbush wrote:

On 3/2/2025 3:01 PM, olcott wrote:

On 3/2/2025 1:27 PM, dbush wrote:

On 3/2/2025 2:21 PM, olcott wrote:

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When formal systems can be defined in such a way that they are not
incomplete and undecidability cannot occur it is stupid to define
them differently.
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That doesn't change the fact that Robinson arithmetic contains the true statement "no number is equal to its successor" that has *only* an infinite connection to the axioms

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If RA is f-cked up then toss it out on its ass.
We damn well know that no natural number is equal to its
successor as a matter of stipulation.

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We know it in RA though *only* an infinite connection to its axioms.
Yet the system still exists, and the axioms of the system make that statement true, but *only* though an infinite connection to its axioms.
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I have eliminated the necessity of systems that contain true statements that have *only* an infinite connection to their truthmakers. All
formal systems that can represent arithmetic do not
contain true statements that have *only* an infinite connection to their truthmakers unless you stupidly define them in a way that
makes them contain true statements that have *only* an infinite connection to their truthmakers.

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As it turns out, any system capable of expressing all of the properties of natural numbers contain at least one true statement that has *only* an infinite connection to its truthmakers.
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Note also that I took the liberty of replacing "incomplete" in your above statement with the accepted definition to make it more clear to all what's being discussed.
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So if you only allow systems where all true statements have a finite connection to their truthmakers, then you don't have natural numbers.
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So choose: do you want to have natural numbers, or do you only want systems where all true statements have a finite connection to their truthmaker?

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Tarski's True(X) is implemented by determining a finite connection
to a truth-maker for every element of the set of human knowledge
and an infinite connection to a truth-maker for all unknowable truths.
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Right, and thus is itself a proxy truth-maker for what it answer.
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Thus given P := ~True(P)
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If True determines that P has no connection to a truth maker, and thus returns false, then P will be true,

 True(LP) determines that P is an infinite sequence,
aborts its evaluation of this infinite sequence
and returns false meaning not true stopping all
evaluation thus not feeding false back into the
evaluation sequence.

But infinite sequences can be true.

 The self-contradictory part of LP is unreachable
in the same way as shown below.

Then True didn't do its job.

 int DD()
{
   int Halt_Status = HHH(DD);
   if (Halt_Status)
     HERE: goto HERE;
   return Halt_Status;
}
 The self-contradictory part of DD emulated by HHH
is unreachable code.
 

There is not "self-contradictory" part of DD, as DD does nothing to be contradicted. DD is HHH-contradictory, and HHH needs to try to figure out how to handle this. HHH can see something that is contradictory to it, and is thus in trouble.
Your problem is you got confused on the identities of the items. DD and HHH are two completely different programs, and defined in a way that HHH must be defined first, as DD has a dependency on HHH, and HHH is the one making the universal claim.
Since DD is created after HHH, it has the ability to contradict HHH if it is smart enough, and since Turing Complete systems allow the embedded of a copy of any program within another, DD can easily "borrow" the "smarts" of HHH to become counter to it, and HHH is just stuck.
Remember, by the time you have DD, you have a SINGLE FIXED HHH that exists, and any talk of some other HHH is pure hypothetical, and doesn't affect DD or what it does, or what the HHH that it is using does.
Since DD is only defined to contradict one particular HHH, the fact that some other variant (which really needs a different name) can get it right doesn't matter, the original HHH, that DD was built on, gets it wrong.