For the relation operator $\leq$,
- Transitive: $\forall x \forall y \forall z; x \leq y\wedge y \leq z \implies x \leq z$
- Reflexive: $\forall x; x \leq x$
- Antisymmetric: $\forall x \forall y; x \leq y \wedge y \leq x \implies x = y$
Now, the axioms of Partial Order is just $\text{Transitive} \wedge \text{Reflexive} \wedge \text{Antisymmetric}$
So, A is FALSE.
For Option B, quoting from Wikipedia:
In general, logical consequence in first-order logic is only semidecidable: if a sentence A logically implies a sentence B then this can be discovered (for example, by searching for a proof until one is found, using some effective, sound, complete proof system). However, if A does not logically imply B, this does not mean that A logically implies the negation of B. There is no effective procedure that, given formulas A and B, always correctly decides whether A logically implies B.
So, B is TRUE.
Axioms of natural numbers are called Peano Axioms. There are $9$ of them and only one of them (axiom of induction) cannot be represented in FOL (requires second order).
So, C is TRUE.
Options D and E are more theoretical and related to fields. If interested you can read here.
Correct option: A
Reference: https://en.wikipedia.org/wiki/First-order_logic