ZFC: Difference between revisions

Line 11:

* 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 controversial 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]].

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]].

@@ Line 11: / Line 11: @@
 * 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 controversial 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]].
+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]].

ZFC: Difference between revisions

Navigation menu

Search