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The set of all mathematical expressions consist of all analytic expressions, closed-form expressions, algebraic expressions, polynomial expressions, and arithmetic expressions.

Do we run into Russell's paradox when talking about a set whose elements are all possible mathematical expressions? or a mathematical expression consisting or equaling all possible mathematical expressions?

If there existed a mathematical expression consisting of all mathematical expressions then it must contain itself and because we can do subtraction on mathematical expressions, would we reach a contradiction?

Similarly, Does there exist a function whose terms are all the functions ever possible?

Cake
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No. A mathematical expression is a finite string of symbols constructed according to certain rules. The symbols and rules vary depending on the system you are working in, but that doesn't change the argument. I have seen it best developed for Peano Arithmetic in the context of Gödel's proof. You can do PA with about a dozen symbols. Gödel defines a mapping from strings of symbols to natural numbers and defines a function that tells you whether a number represents a string of symbols that is a legal sentence according to the rules of PA. You can't have a mathematical expression that consists of all mathematical expressions because the first has to be finite and the second is countably infinite. In PA you can't (easily) talk about sets, but in ZFC you can. ZFC can define the set of Gödel numbers of legal ZFC mathematical expressions. It is a countably infinite set, but there is no reason it should contain itself so there is no Russell's paradox.

Ross Millikan
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