Syntactic derivations in second-order logic based on the Comprehension Axiom Schema and Axioms of Choice are very much like syntactic derivations in set theory, and working mathematicians write both in shorthand
A formal system that goes beyond basic logic by allowing you to quantify over (talk about) properties and relations themselves, not just individual objects.
Set theory(as used in mathematics)
A branch of mathematics that studies collections of objects (called 'sets') and the rules for how they relate to each other.
Shorthand(in mathematical and logical notation)
A simplified or abbreviated way of writing something that experts understand, but which leaves out details that experts don't need spelled out.
Syntactic derivations(in logic and mathematics)
A step-by-step logical proof where you start with basic rules and combine them to reach a new conclusion, following strict grammatical rules like you would in a language.
Let \(c,d\in (a,b)\) such that \(f(c)<0\) and \(f(d)>0\). Without loss of generality, \(c<d\). Let \(X=\{e\in(a,b) : f(e)<0\}\). Since we have relation variables for subsets of the domain, we can think of X simply as a value of such a relation variable. , \(X=\{e : e\notin X\}\)) and then we should not be able to claim that it exists. However, in this case the Comprehension Axiom Schema implies that X exists. Clearly, \(X\ne\emptyset\) and X is bounded from above by d. One of the sec