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== Weakly inaccessible ==
== Weakly inaccessible ==
Weakly inaccessible cardinals were first introduced by Hausdorff in an attempt to resolve the [[continuum hypothesis]].<ref>Hausdorff, Felix. "[https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:%5b453%5d} Grundzüge einer Theorie der geordneten Mengen]", Mathematische Annalen, vol. 65, num. 4 (1908), pp.435--505. DOI:10.1007/BF01451165.</ref> Cardinals \( \aleph_\alpha \) for limit ordinal \( \alpha \) are known as limit cardinals, since applying the cardinal successor operator to a cardinal less than \( \aleph_\alpha \) yields a cardinal also less than \( \aleph_\alpha \). (citation for this being the etymology? {{citation needed}}) However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, these cardinals are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, for cardinals such as \( \aleph_1 \), the limit of any countable sequence of countable ordinals is countable, so no system of normal functions can build up a sequence shorter than \( \aleph_1 \) cofinal in \( \aleph_1 \) - thus \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the length-\( \omega \) sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well.
Weakly inaccessible cardinals were first introduced by Hausdorff in an attempt to resolve the [[continuum hypothesis]].<ref>Hausdorff, Felix. "[https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:%5b453%5d} Grundzüge einer Theorie der geordneten Mengen]", Mathematische Annalen, vol. 65, num. 4 (1908), pp.435--505. DOI:10.1007/BF01451165.</ref> Cardinals \( \aleph_\alpha \) for limit ordinal \( \alpha \) are known as limit cardinals<ref>Jech, Thomas (2003), ''Set Theory'', Springer Monographs in Mathematics (third millennium ed.), Berlin, New York: Springer-Verlag</ref>, since applying the cardinal successor operator to a cardinal less than \( \aleph_\alpha \) yields a cardinal also less than \( \aleph_\alpha \). However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, these cardinals are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, for cardinals such as \( \aleph_1 \), the limit of any countable sequence of countable ordinals is countable, so \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the length-\( \omega \) sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well.


The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega \) acts as a suitable diagonalizer in an ordinal collapsing function such as Madore's or Bachmann's.
The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega \) acts as a suitable diagonalizer in an ordinal collapsing function such as Madore's or Bachmann's.
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Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming.
Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming.


Notice that if the uncountability requirement is dropped, \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit of any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). Zermelo referred to the strongly inaccessible cardinals including \( \aleph_0 \) as "Grenzzahlen".<ref>A. Kanamori, "[https://math.bu.edu/people/aki/10.pdf Zermelo and Set Theory]", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.</ref> However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it similarly would be weakly inaccessible, although vacuously. For this reason many authors add the condition of uncountability. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite" ({{citation needed}}, Jonsson cardinals may be small but they have high consistency strength, and they're considered the higher infinite)
Notice that if the uncountability requirement is dropped, \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit of any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). Zermelo referred to the strongly inaccessible cardinals including \( \aleph_0 \) as "Grenzzahlen".<ref>A. Kanamori, "[https://math.bu.edu/people/aki/10.pdf Zermelo and Set Theory]", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.</ref> However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it similarly would be weakly inaccessible, although vacuously. For this reason many authors add the condition of uncountability. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite".


You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means.<ref>Resurrection axiom paper</ref>
You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means.<ref>Resurrection axiom paper</ref>
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