first-order

Ultimately, the issue is that the first-order theory of real closed fields contains no axiom approaching the induction scheme of PA in power.

In one meaning, the first-order logic is called complete because every logically valid statement is provable from the logical axioms.

A lot of the GoF patterns are mainly useful in statically typed languages and/or languages lacking first-order functions or closures.

Second-order systems contain statements that first-order systems can't express at all, much less prove.

This kind of statement is invalid in first-order logic, for some good reasons.

The first-order ones like we're used to seeing are popular because they're simple to understand and implement not because they're the only way.

If this is the case, this is like applying a first-order splitting method in which turns are handled with the forward Euler method, which would be quite inaccurate.

The fact that we have a Goedel's Completeness Theorem is really what allows us to confidently use first-order logic as a conceptual basis for all mathematics.

The system need be no more complex than first-order Peano arithmetic, which is the Peano axioms (reflexivity, transitivity, succession, etc.) combined with the concept of induction.

Also, since the Chapman-Enskog method is asymptotic, it is not guaranteed that higher-order equations (inviscid Euler equations being the zero-order equations and Navier-Stokes equations being the first-order equations) will provide an accurate description of flows.

Goedel's Completeness Theorem really says something important about the usual bunch of logical axioms and inference rules, the one that form the backbone of pretty much every first-order axiomatic system we study.

For Java, you get some very excellent tooling, a slowly evolving language, a battle-tested and mature virtual machine, comprehensive API, first-order parametric polymorphism, and a number of useful frameworks with tons of community support.

It tells us that axiomatic systems for first-order logic are complete in the sense of "strong enough to prove all tautologies".

That's one meaning of "complete", and it's not changing from one axiomatic system to another, as long as we stay in first-order logic.

This is why when you say "The Incompleteness Theorem shows that no first order axiomatic system is powerful enough to encapsulate all true statements in the second order Peano axioms", this is nonsense, because a first-order axiomatic system cannot prove statements in a second-order system, in general.

In fact, if first-order logic were incomplete (say if you removed a crucial logical axiom from it) and the Completeness Theorem wouldn't hold, it would still be true that you could take all true statements about the integers, call them axioms, and get a complete axiomatic system in the other meaning of "complete".