I was watching this video about transfinite numbers and someone commented this. I swear I can't be alone in thinking that "First Inaccessible Cardinal: Omega Fixed Point" sounds like some kind of ultimate technique in a math-based anime or whatever.
(The video, for reference:)
It's worth noting that the commenter is actually incorrect.
Not in that an inaccessible is an omega fixed point, every inaccessible is provably an omega fixed point (or aleph fixed point, these are used interchangeably since for any ordinal number ω_α, it's cardinality is exactly א_α), which is an ordinal α such that ω_α = α. It is fixed under the omega assignment function.
But rather in the (implicit) assertion that there is only one limit to the aleph function, and that it is inaccessible. Neither of which are true. There are many limits to the aleph function, infinitely many. The smallest of these is generally given the name "Omega Fixed Point" despite all others also being omega fixed points, and is also not inaccessible.
The issue arises in a property called "cofinality". The cofinality of an ordinal A is the smallest size of a set of ordinals smaller than A such that for every single ordinal smaller than A, there exists an element of the set which is bigger than it.
In terms of our observations, this amounts to the smallest sequence of ordinals that can be moved through and will eventually reach A. The cofinality of an inaccesible cardinal is itself, by definition, but the cofinality of the Omega Fixed Point (I put it in capitals to differentiate it from other omega fixed points) is exactly omega, or aleph 0, or countable infinity, which is much smaller than Omega Fixed Point. (Define a sequence as follows: begin with ω, then ω_ω, ω_ω_ω, and so on. This sequence if continued forever reaches the Omega Fixed Point.)
The reason we call an inaccessible cardinal inaccessible is because it cannot be reached from below using the usual operations of powerset, union, succession, etc. We must instead assert it's existence, much like infinity itself. What's strange is that this time, the cardinal is so big that ZFC itself cannot assert the existence of one, because to do so would be to admit a model exists for ZFC, which would prove ZFC consistent, which is not allowed as proven by Gödel so long ago.
