Completeness of Logic versus completeness of a theory

Question

It's my understanding that First-Order-Logic is complete, but that not all logical theories are complete (for example, Russel's Arithmetic) - both these results having been shown by Godel.

I'm trying to clarify my understanding of these facts. Can they be restated in the following terms? :

Suppose a conjunction $C_1$ of statements in First-Order-Logic 'forces' the statement $S_1$ to be true. In other words, in all models where $C_1$ are true, $S_1$ is true as well. In other words, $(C_1\ \&\ \neg S_1)$ is unsatisfiable.

Given the completeness of First-Order-Logic, we know :

$\vdash (C1 \rightarrow S1) $

In other words, $(C_1 \rightarrow S_1)$ is a logical tautology.

Given that not all logical theories are complete, the following may not necessarily be true :

$C_1 \vdash S_1$

Have I gotten this right?

Thanks!

The two meaning of "completeness" are no the same: (1) Completeness of FOL means that $\Gamma \vdash S \text { iff } \Gamma \vDash A$. — Mauro ALLEGRANZA, Jan 05 '20 at 17:29
To say (2) that $\mathsf {PA}$ is "incomplete" means that there is a sentence $S$ such that neither $\mathsf {PA} \vdash S$ nor $\mathsf {PA} \vdash \lnot S$. — Mauro ALLEGRANZA, Jan 05 '20 at 17:30
Applying FOL Completeness to the FOL theory $\mathsf {PA}$ means that there is a model $\mathcal M$ of $\mathsf {PA}$ such that $\mathcal M \nvDash S$ and a model $\mathcal M'$ of $\mathsf {PA}$ such that $\mathcal M' \nvDash \lnot S$. — Mauro ALLEGRANZA, Jan 05 '20 at 17:32
The "completeness" of (2) is meant with respect to the "standard" model $\mathbb N$ of $\mathsf {PA}$. — Mauro ALLEGRANZA, Jan 05 '20 at 17:33
See the post what-is-the-difference-between-Godel-completeness-and-incompleteness-theorems — Mauro ALLEGRANZA, Jan 05 '20 at 17:35
This is a heck of a lot clearer to me now @mauro-allegranza . Thank you so much!! — user1050268, Jan 05 '20 at 17:43

Completeness of Logic versus completeness of a theory

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