Meeting point of the three hierarchies

The meeting point of the three hierarchies is equal to \(f_{\theta(\Omega_{\omega})}(100)\) using the fast-growing hierarchy and Feferman's \(\theta\)-function with this system of fundamental sequences.Hayden's Big Numbers. Retrieved 2022-05-16. Also, the creator insisted that this number is equal to \( f_{\psi_0(\psi_{\omega}^3(0))}(100)\) using the fast-growing hierarchy with fundamental sequences using equalities to outputs of Buchholz's function, but later it no longer does.Hayden's Big Numbers. Retrieved 2022-07-11. The term was coined by googology user HaydenTheGoogologist2009.
== Etymology ==
According to the creator, the number was named after a statement that \(\theta(\Omega_{\omega})\) is the least "catching ordinal", which is cited from a blog post by Googology Wiki user hyp cos. However, the statement and the blog post have many problems:
# It uses an unspecified (and perhaps ill-defined) OCF, and is supposed to be different from Feferman's \(\theta\).
# It does not use an specific system of fundamental sequences, and hence the fast-growing hierarchy and the slow-growing hierarchy in the analysis is ill-defined.
# Even if one fixes a system of fundamental sequences, there is no proof of the comparison, and even the comparability in the context of the "catching" property is not appropriately formalised. (Several Googology Wiki users pointed out the informality and the ill-definedness of the catching propertya comment by LittlePeng9a comment by Vel!, and asked a proof for results (possibly on an alternative formulation)a comment by Deedlit11a comment by Vel!. However, no actual proof is explained.)
Therefore the "catching" property in the cited source does not mathematically make sense, and is dubious even if one fixes an alternative formulation.
== Approximations ==
== Sources ==
Category:Numbers
Category:Numbers by HaydenTheGoogologist2009
Category:Higher computable level