If one were to verify that
$$ \sqrt{2} < 3 $$
would the underlying formalisation require a logic more expressive than first-order? Or, is FOL sufficient since real numbers can be formalised in set theory?
How about evaluating differentials, like
$$ \frac{d}{dy} (3x+2) $$
what how expressive does the logic need to be?
Thanks
Comparing particular algebraic numbers (localized away from their Galois conjugates by inequalities, such as the positive root of $X^2=2$) can be done in any theory that supports arithmetic calculation. For example, the first-order theory of real-closed fields is decidable and that includes decisions about statements of the form $a > b$ comparing two real algebraic numbers.
Model theory of differential (and difference) fields and rings usually takes place in a first-order logic context similar to fields. Theorems are proved by second-order methods but the object of study is a first-order theory.