>>10972303>And Gödel showed that what is possible to derive is less than what's true within that constructed system (any system sufficient to sustain the natural numbers). The issue with your phrasing is that you pretty much leave open what "true" means.
>If truth within the system is exactly the same as what can be derived, we call this system complete.If you take a proposition [/math] P [/math] being "true" to mean provable in a theory
then you're just adopting a Brouwerian stance to truth.
In any case, we call a (syntactic) system complete if every statement which is (semantically) true in all models of the system is also a theorem of its (purely syntactic) calculus. I.e. (semantically) complete means
The word "complete" in "in-completeness theorem" however refers to decidability or .
Gödels proof of the in-completeness theorem is syntactic and even intuitionistic and shows that there are undecidable statements.
>Some systems are complete, for example prepositional logic.Yeah, Gödels completeness theorem proves first order propositional logic (FOL) complete. Arithemtic can be written down in FOL.
As a consequence all statements true in all models of arithmetic are provable.
Arithmetic surely has a model, and e.g. the set theory models
0:={}, 1:={0}, 2:={1}, ...
Any arithmetic model also induces many isomorphic models. E.g. if
0, 1, 2, 3, 4, 5,... with the successor operation S mapping x to x+1 has a model, then
0, 2, 4, 6, 8, 10,... with the successor operation T mapping x to x+2 is also a model. One in which "6" is "3", but after you know the identification, it describes the same things and relation.
Since FOL is complete and that arithmetic has undecidable propositions, we know that arithmetic doesn't have a unique model.
"Truth" is barely affected.
