ZFC

ZFC (Zermelo-Fraenkel with choice) is the most common axiomatic system for set theory, which provides a list of 9 basic assumptions of the set-theoretic universe, sufficient to prove everything in mainstream mathematics, as well as being able to carry out ordinal-analyses of weaker systems such as KP and Z2. The axioms are the following:

• Axiom of extensionality: two sets are the same if and only if they have the same elements.
• Axiom of regularity: for all \(x\), if \(x \neq \emptyset\), then there is \(y \in x\) so that \(y \cap x = \emptyset\).
• Axiom of separation: Given any set \(X\) and any formula \(\varphi(x)\), \(\{x \in X: \varphi(x)\}\) is also a set.
• Axiom of pairing: If \(x\), \(y\) are sets, then so is \(\{x, y\}\).
• Axiom of union: For any set \(x\), there is a set \(y\) such that the elements of \(y\) are precisely the elements of the elements of \(x\).
• Axiom of replacement: For all \(X\), if \(\varphi(x,y)\) is a formula so that \(\forall x \in X \exists! y \varphi(x,y)\), then there is some \(Y\) so that \(\forall x \in X \exists y \in Y \varphi(x,y)\).
• Axiom of infinity: there is an inductive set.
• Axiom of powerset: Given any set \(x\), \(\{X: X \subseteq x\}\) is also a set.
• Axiom of choice: Given any set \(X\), there is a function \(f\) so that, for all nonempty \(x \in X\), \(f(x) \in x\).

ZF denotes the theory of ZFC, minus the axiom of choice, which is a controversial axiom due to consequences such as the Banach-Tarski paradox. However, ZF also has its own flaws, such as not being able to prove every set has a well-ordering (which is equivalent to the axiom of choice) and not being able to do cardinal arithmetic or even prove cardinals are comparable. \(\mathrm{ZFC}^-\) or \(\mathrm{ZF}^-\) denote the even weaker theories of ZFC or ZF, respectively, minus the axiom of powerset. These both have the same strength as full Z2. The even weaker theory of \(\mathrm{ZFC}^{--}\)^{[Citation needed]}, where separation is restricted to \(\Delta_0\)-formulae, has the same strength as KP.

Gödel's incompleteness theorems guarantee that there are sentences not provable or disprovable in ZFC, if it is consistent. This incompleteness phenomenon is surprisingly pervasive, and includes sentences such as the axiom of constructibility \(V = L\), the continuum hypothesis, the generalized continuum hypothesis, the diamond principle, or the existence of a weakly inaccessible cardinal.