Ordinal collapsing

Ordinal collapsing is a process by which ordinals are reduced in size, most frequently in the context of ordinal collapsing functions (OCFs) or indexes in ordinal hierarchies. While the process is slightly different for each notation, generally collapse is made possible by the following:

• If collapse of a successor ordinal is needed in a structure, decrement the ordinal by one and change the structure in some way. In the slow-growing hierarchy (SGH) one decrements the index and adds one to the expression whereas in the fast growing hierarchy (FGH) one decrements the index by one and iterates the expression.
• If collapse of a limit ordinal with cofinality ω is needed in a structure, reduce the ordinal to the nth term in its preassigned fundamental sequence and change the structure in some way. Sometimes this may mean not changing the structure at all, as in the case of the SGH and FGH which both reduce the ordinal and leave everything else unchanged.
• If collapse of a regular ordinal α > ω is needed in a structure, reduce the ordinal to the nth term in the largest possible fundamental sequence whose limit is beneath α. Because notations that use regular collapse (usually OCFs) are so far-reaching, generative expansion techniques are used which take the "largest possible fundamental sequence" to be the largest possible at that moment in the expansion, and the means of generating this adds subtlety to the OCF in question as with, for example, Ytosk's 2-shifted OCF opposed to traditional Buchholz, the former having more power than the latter due to the exact expansion of regular ordinals used.
• If collapse of a regular ordinal α whose cofinality β is such that ω < β < α, expand in the part of the ordinal indexed by β and use the regular collapsing technique in the previous bullet point. For example, in most OCFs Ω_Ω is collapsed by noting cof(Ω_Ω) = Ω and expanding into the Ω index with the largest possible countable ordinal according to the context of the expression that Ω_Ω is in.

Collapsing Ω

Ω is the first real challenge in ordinal collapsing, because Ω is regular and has no countable fundamental sequence. The way this is usually dealt with is by taking a portion of the OCF that only outputs countable ordinals and setting the "fundamental sequence" of Ω to be the sequence of current largest countable ordinals. To make an OCF a proper function (that is to say, there is only one output for a given input), the last ψ₀ (simply called ψ) and the entire expression except for the Ω being expanded into is iterated. Here are some examples:

ψ(Ω)
 = ψ(ψ(ψ(...)))
ψ(Ω2)
 = ψ(Ω + Ω)
 = ψ(Ω + ψ(Ω + ψ(Ω + ...)))
ψ(Ω2)
 = ψ(Ω•Ω)
 = ψ(Ω•ψ(Ω•ψ(Ω•...)))