Re: The philosophy of logic reformulates existing ideas on a new basis ---

On 11/8/24 1:08 PM, olcott wrote:

On 11/8/2024 12:02 PM, Richard Damon wrote:

On 11/8/24 12:25 PM, olcott wrote:
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That formal systems that only apply truth preserving
operations to expressions of their formal language
that have been stipulated to be true cannot possibly
be undecidable is proven to be over-your-head on the
basis that you have no actual reasoning as a rebuttal.
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No, all you have done is shown that you don't undertstand what you are talking about.
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Godel PROVED that the FORMAL SYSTEM that his proof started in, is unable to PROVE that the statement G, being "that no Natural Number g, that satifies a particularly designed Primitive Recursive Relationship" is true, but also shows (using the Meta-Mathematics that derived the PRR for the original Formal System) that no such number can exist.
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 The equivocation of switching formal systems from PA to meta-math.
 

No, it just shows you don't understand how meta-systems work.
The Formal System is PA. that defines the basic axioms that are to be used to establish the truth of the statement and where to attempt the proof of it.
The Meta-Math, is an EXTENSION to PA, where we add a number of additional axioms, none that contradict any of the axions of PA, but in particular, assign each axiom and needed proven statement in PA to a prime number. These provide the additional semantics in the Meta-Math to understand the new meaning that a number could have, and with that semantics, using just the mathematics of PA, the PRR is derived that with the semantics of the MM becomes a proof-checker.
Note, the Meta-Math is carefully constructed so that there is a correlation of truth, such that anything true in PA is true in MM, and anything statement shown in MM to be true, that doesn't use the additional terms defined, is also true in PA.
There is no equivocation in that, as nothing changed meaning, only some things that didn't have a semantic meaning (like a number) now does.
If you want to try to show an actual error or equivocation, go ahead and try, but so far, all you have done is shown that you don't even seem to understand what a Formal System is, since you keep on wanting to "re-invent them" but just repeat the basic definition of them.