ZFC: Difference between revisions
May sound like ZF proves Banach-Tarski
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(May sound like ZF proves Banach-Tarski) |
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* 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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