# Computing square roots and calculus

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?

$$\frac{d}{dy} (3x+2)$$

what how expressive does the logic need to be?

Thanks

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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.