>all the brainlets itt arguing metalogic without knowing what model theory even isshiggy
>>13750498>If you can't prove something, then it's not trueAbsolutely wrong and retarded. Even a child could look at the usual first order Peano axioms and identify each and every axiom as being something obviously true. This is all basic shit like, "for every natural number, there's another that comes after it" and "zero doesn't come after any natural number". Any system containing only these axioms and basic logical axioms (modus ponens, etc.) HAS to be consistent, or the entirety of mathematics would be bullshit, at least as a tool for modeling physical systems, under the usual assumptions about what constitutes physical "consistency". Empirically, this has shown not to be the case, so for all practical purposes, such a theory must be logically consistent, right? So we take the formula representing the theory's consistency as being "true" a priori. You can always construct such a first-order formula, if the theory constitutes some fragment of natural arithmetic.
Yet, the (second) Incompleteness Theorem prohibits us from actually proving this consistency result using only the original (first-order) theory. This is a metalogical result based on both the structure of first order logic and the structure of "models", in the sense of model theory. We might be able to prove consistency in a stronger theory with more axioms, but then we will be unable to prove our new "theory-proving" theory, without adding additional axioms, and so on and so forth. But none of this changes the fact that the original Peano axioms (theory of the natural numbers), as well as the logical axioms, are known to be true a priori, yet we aren't able to prove it.
It turns out that there are OTHER, more "basic" things that we can show (using model theory, or other tools) are true of the natural numbers, that we are also unable to prove. This is the main result of the (first) Incompleteness Theorem.
