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

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?