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.
<s>Both varieties of inaccessible cardinal were introduced to generalize how unreachable \( \omega \) is from the finite numbers.</s> Weakly inaccessible cardinals were introduced by Hausdorff to try to work on the continuum hypothesis.
Both varieties of inaccessible cardinal were introduced to generalize how unreachable \( \omega \) is from the finite numbers. Notice that \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit fo 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 is vacuously. weakly inaccessible. Therefore, many authors add the condition of uncountabiltiy. 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"▼
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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.▼
▲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>
== Grothendieck universes and categoricity ==
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