Précis of epistemology/ZFC is false

For a theory to be false, it suffices that only one of its axioms is false.

Zermelo's axiom of separation: any well-defined formula determines a part of a set that contains all of its elements for which it is true, and only them.

This axiom is true by definition of a well-defined formula: a formula is well-defined when its truth, or its falsity, is determined in all cases where it is applied.

Zermelo formulated his axiom being careful not to specify which formulas are well defined, because he did not know. To formalize Zermelo's theory, Fraenkel completed it with the following principle, suggested by Skolem: all the formulas of a pure set theory, stated with first-order logic, are well defined. But this principle is false, because these formulas can contain expressions such as "for any set" or "there is a set". However, ZFC does not precisely determine the concept of set. What is a set? What is not a set? The universe of all sets can be defined in many different ways. The truth of a formula which contains "for any set" may depend on the chosen interpretation. These formulas are therefore not always well defined. Hence, Fraenkel's formulation of Zermelo's axiom of separation is false. ZFC is therefore false.

For example, ZFC makes it possible to prove the existence of the set equal to {} if the axiom of constructibility is true and to {{}} if it is false. Since the truth of the axiom of constructibility depends on the definition of the universe of all sets, this set, which ZFC allows to prove the existence, is not well defined. A set that is not well defined is not a set at all. It does not exist. ZFC allows to prove false theorems, therefore it is false.

To correct Fraenkel's error, it suffices to prohibit unbounded quantifiers in the definition of sets. For the replacement axiom, we need a theory of well-defined functional relations.

ZFC is the standard theory adopted until now, for more than 70 years, by almost all mathematicians in the world, as the foundation of mathematical knowledge.