Inaccessible cardinal: Difference between revisions
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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.
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
▲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 \( 2^n \). 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 accountability. 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)
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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