ZFC: Difference between revisions
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* 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 [[Second-order arithmetic|Z2]]. The even weaker theory of \(\mathrm{ZFC}^{--}\){{citation needed}}, where separation is restricted to \(\Delta_0\)-formulae, has the same strength as [[Kripke-Platek set theory|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 [[Constructible hierarchy|axiom of constructibility]] \(V = L\), the continuum hypothesis, the generalized continuum hypothesis, the diamond principle, or the existence of a [[Inaccessible cardinal|weakly inaccessible cardinal]].
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