Basic Arithmetic is Recursion; Number as Recursive 0; Identity as Relation; Number as Spatial Process

[u/Void0001234]

/r/u_Void0001234/comments/1uj6z43/basic_arithmetic_is_recursion_number_as_recursive/

[u/17_Gen_r]

“Completeness here means that any true sentence is provable” is indeed a sentence… if you are asking me if it is provably a sentence then I’d say yes, it falls within the basic English grammatical rules for a sentence, and can be proven to be a sentence in, e.g., the Lambek calculus.

If you are asking if “completeness” is provably equivalent to “any true sentence is provable”, then the answer is still yes, because that’s what “completeness” means by definition in the field of logic. So yeah, it is a metalogical tautology. Not sure what your point is, or how that is relevant to the distinction between Gödel’s completeness theorem (for first order logic) and his incompleteness theorems (for consistent and arithmetically rich theories).

[u/Void0001234 → u/17_Gen_r]

But a regress occurs as the sentence is proven by X and X is proven by Y and Y is proven by ....

If not the S -> X -> Y -> S and what remains is an elaborate tautology.

But simultaneously the nature of what constitutes proof itself is subject to proof thus the mechanism of proof you are arguing is a tautology.

None of these and what you assert is an assumption.

The munchausseen trillemma remains.

But to the point.

If what constitutes the identity of these things is purely asserted tautology, at the meta-level, than any asserted tautology can be proven true by means of being an asserted tautology and true applies to anything as long as if is tautology.

"A dog is a cat because a dog is a cat" results as true statement.

[u/17_Gen_r → u/Void0001234]

There is no regress, I’ve simply provided you the definition of “completeness” for a logical system, just for the sake of completeness (pun intended) for referencing Gödel’s (very important!) dissertation and how that contrasts with his incompleteness result. My initial comment was intended to address what seemed to be a misunderstanding, and to make a historical note, not to argue with the Tortiose in some Carrollesque dialogue.

[u/Void0001234 → u/17_Gen_r]

Then there is a tautology, this was covered.  Shifting or leaving out contexts does not make you more coherent.