If you write out your statements encoded in say ZFC no one would recognize what you are saying. Sometimes the concept of sets and set relationships fit our needs exactly. Other times we build more user friendly concepts on top of set theory like an API. Categorial constructions are also everywhere whether one recognizes them or not.
Call a real number "algebraic" if it's a zero to some polynomial with rational coefficients. (e.g. \sqrt{2} is algebraic since it's a zero for x^2 - 2). Claim: There exist non-algebraic ("transcendental") numbers. Proof: There are only countably many polynomials, and so there are only countably many algebraic numbers, but there are uncountably many reals. Similarly, there are numbers that aren't Turning-computable. Etc.