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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception ---
 Ultimate Foundation of Truth
Date: Sun, 2 Mar 2025 16:25:03 -0500
Organization: i2pn2 (i2pn.org)
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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:
>>>>>
>>>>> 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.
>>>>>
>>>>
>>>> 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
>>>
>>> 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.
>>
>> 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.
>>
>>>
>>> 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.
>>
>> 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.
>>
>> 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.
>>
>> So if you only allow systems where all true statements have a finite 
>> connection to their truthmakers, then you don't have natural numbers.
>>
>> 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?
> 
> 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.
> 
>

Right, and thus is itself a proxy truth-maker for what it answer.

Thus given P := ~True(P)

If True determines that P has no connection to a truth maker, and thus 
returns false, then P will be true, and thus shows that True has made an 
error, as the expression P HAS a connection, a finite one in fact, to 
its proxy truth-maker of True.

This is not allowed.

If True determines that P has that connection shown above, then P will 
be false, and thus we find that True has declared a false statement to 
have a truth maker.

This is not allowed either.

Thus, True can not exist.

The problem is that P defined as ~True(P) is an expression that can be 
created to exist in the Theory, based on the Meta-Theory created, as 
long as the Theory can express the properties of the Natural Numbers, 
and has a True Predicate.

Thus, The expression can't just be "rejected" because it was a valid 
statement.

The answer is that such a True Predicate can't exist.