Aleph fixed point: Difference between revisions
RhubarbJayde (talk | contribs) (Created page with "An aleph fixed point, also referred to as an omega fixed point (OFP), is a fixed point of the function \(f(\alpha) = \aleph_\alpha\). In other words, it is a cardinal \(\kappa\) so that \(\aleph_\kappa = \kappa\). The existence of such a \(\kappa\) is guaranteed by the axioms of infinity, powerset and replacement combined with Veblen's fixed point lemma, and therefore it is provable in ZFC. Aleph fixed points are large in that they are unreachable from below via the...") |
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An aleph fixed point, also referred to as an omega fixed point (OFP), is a fixed point of the function \(f(\alpha) = \aleph_\alpha\). In other words, it is a cardinal \(\kappa\) so that \(\aleph_\kappa = \kappa\). The existence of such a \(\kappa\) is guaranteed by the axioms of infinity, powerset and replacement combined with Veblen's fixed point lemma, and therefore it is provable in [[ZFC]]. Aleph fixed points are large in that they are unreachable from below via the aleph operator. However, it is possible that the number of real numbers is an aleph fixed point, or more. Furthermore, the least aleph fixed point has cofinality \(\omega\), which follows from the [[Normal function|normality]] of \(f(\alpha) = \aleph_\alpha\). A regular aleph fixed point is precisely a [[Inaccessible cardinal|weakly inaccessible cardinal]], and, therefore, the [[large cardinal]] hierarchy is beyond the notion of aleph fixed points, the fixed points of their enumeration, and so on, since those can all be proven to exist and are less than the least weakly inaccessible cardinal, if it exists. |
An aleph fixed point, also referred to as an omega fixed point (OFP), is a fixed point of the function \(f(\alpha) = \aleph_\alpha\). In other words, it is a cardinal \(\kappa\) so that \(\aleph_\kappa = \kappa\). The existence of such a \(\kappa\) is guaranteed by the axioms of infinity, powerset and replacement combined with Veblen's fixed point lemma, and therefore it is provable in [[ZFC]]. Aleph fixed points are large in that they are unreachable from below via the aleph operator. However, it is possible that the number of real numbers is an aleph fixed point, or more. Furthermore, the least aleph fixed point has cofinality \(\omega\), which follows from the [[Normal function|normality]] of \(f(\alpha) = \aleph_\alpha\). A regular aleph fixed point is precisely a [[Inaccessible cardinal|weakly inaccessible cardinal]], and, therefore, the [[large cardinal]] hierarchy is beyond the notion of aleph fixed points, the fixed points of their enumeration, and so on, since those can all be proven to exist and are less than the least weakly inaccessible cardinal, if it exists. |
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Revision as of 05:53, 25 March 2024
An aleph fixed point, also referred to as an omega fixed point (OFP), is a fixed point of the function \(f(\alpha) = \aleph_\alpha\). In other words, it is a cardinal \(\kappa\) so that \(\aleph_\kappa = \kappa\). The existence of such a \(\kappa\) is guaranteed by the axioms of infinity, powerset and replacement combined with Veblen's fixed point lemma, and therefore it is provable in ZFC. Aleph fixed points are large in that they are unreachable from below via the aleph operator. However, it is possible that the number of real numbers is an aleph fixed point, or more. Furthermore, the least aleph fixed point has cofinality \(\omega\), which follows from the normality of \(f(\alpha) = \aleph_\alpha\). A regular aleph fixed point is precisely a weakly inaccessible cardinal, and, therefore, the large cardinal hierarchy is beyond the notion of aleph fixed points, the fixed points of their enumeration, and so on, since those can all be proven to exist and are less than the least weakly inaccessible cardinal, if it exists.
In most if not all OCFs, the collapse of the least aleph fixed point is the Extended Buchholz ordinal, which is why it is sometimes alternately referred to as the OFP, although this is technically a misnomer.