Inaccessible cardinal: Difference between revisions

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"

 

 

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.

A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent.