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

On 3/2/25 8:05 PM, olcott wrote:

On 3/2/2025 6:42 PM, Richard Damon wrote:

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,

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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.

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But infinite sequences can be true.
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Proving the Goldbach has a finite proof for each element
of the infinite set of natural numbers thus makes progress
towards its goal.
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The evaluation of the Liar Paradox gets stuck in an infinite
loop and never makes any progress towards resolution.
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Clocksin and Mellish understood this. You are so sure that
I must be wrong that you did not bother to see that they
understood this.
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 I did read it, and clearly they don't understand what Godel's G is, as it does not have infinite recursion in it.