Inaccessible cardinal: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first invented. Essentially, cardinals such as \( \al...") |
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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"
You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\
Due to this difficulty and sufficiency of weakly inaccessible cardinals, as previously mentioned, strongly inaccessible cardinals are rarely used in apeirological circles. However, they are quite commonly used in the literature, due to results in the next section.
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