User blog comment:Nayuta Ito/faketest/e0/@comment-31966679-20180809132507/@comment-30754445-20180809142951

Putting it in more layman-friendly terms:

(1) Ω, I, M and K are ordinals.

(2) Ω < I < M < K.

(3) Ω is the smallest "uncountable ordinal". The technical definition doesn't matter. What matters in terms of googology is this: No sequence of ordinals smaller than Ω will have Ω as its limit. If you got what I meant by this, skip to (5). Otherwise proceed to (4).

(4) ω, as you already know, is the limit of 1,2,3,4,... . Similiarly, ε0 is the limit of ω,ωω,ωω ω ... . The ordinals you are familiar with can all be described as either:

(a) another ordinal + 1

-or-

(b) A limit of sequence of smaller ordinals, as demonstrated above.

So Ω is the smallest ordinal for which this cannot be done. That's why it is called "an uncountable ordinal".

(5) Since Ω is not the limit of any sequence, we cannot use it directly to create large numbers. Writing fΩ(n) makes no sense, because we cannot speak of "the n-th ordinal in Ω's fundamental sequence".

(6) To bypass this problem, we use the ψ's. What ψ does is this: It takes an ordinals (which can be uncountable) and creates a countable ordinal from them.

This is called "ordinal collapse".

For example, in UNOCF, ψ(Ω)=ε0. In this case, ψ took the uncountable ordinal Ω and transformed it (collapsed it) into the ordinary countable ordinal ε0.

(7) You might wonder why bother writing ψ(Ω) when you could simply write ε0. Well, as it turns out, using this trick with ψ enables us to generate countable ordinals which are far bigger than anything we could create otherwise.

A few examples:

(a) φ(ω,0) = ψ(Ωω) in UNOCF.

(b) The limit of what you can create with φ(a,b) = ψ(ΩΩ) in UNOCF.

(c) You can put more variables in the φ to make it stronger. For example, the limit of φ(a,b,c) = ψ(ΩΩ 2 ).

(d) The limit of φ(a,b,c,d) = ψ(ΩΩ 3 ).

(e) The limit of φ(a,b,c,...,n) with any number of variables = ψ(ΩΩ ω ). This ordinal is also called SVO (Small Veblen Ordinal)

(f) You can actually go a bit beyond that with φ, by allowing a number of variables which, in itself, is an ordinal! So instead of φ(1,0,0,0) = φ(1 in the 4th position), you can have φ(1 in the ωth position) or φ(1 in the ε₀th position) or even φ(1 in the SVOth position)!

The limit of all that is called LVO (Large Veblen Ordinal), and in UNOCF you can write it as ψ(ΩΩ Ω )

(g) Beyond this, the extended Veblen functions break down, but with ψ we can go further to ψ(ΩΩ Ω Ω ) and ψ(ΩΩ Ω Ω Ω   ) and so on.

For an infinite tower of Ω's, we get the ordinal known as the BHO (Bachman-Howard Ordinal).

(8) Beyond that, we need more ψ functions. Just like we used ψ to create ordinary ordinals from Ω's, we now need a "super-uncountable ordinal" (Ω2) and a "super-ψ" function (ψ1) that collapses super-uncountable ordinals to ordinary uncountable ordinals.

Note that UNOCF uses these concepts in quite a different way than the usual way of doing things. But the details don't really matter. I'm just giving you a quick overview.

(9) We can add more and more levels similar to the outline illustrated above. How many? Well, as many as we wish! So we start to count them:

Ω3, Ω4, ..., Ωω , ... ΩΩ, ...

And of-course, for each one of these there's also an accompaning ψ-function that collapses it into a lower order.

(10) We can continue the above until we reach an Ordinal a, such that a = Ωa. Then we're stuck, if we're going to stick with ordinary collapse functions.

(11) To get out of this rut, we introduce I. The exact definitions are, again, unimportant at this time. Suffice to say that with I (and a new set of collapse functions) we can bypass all those pesky troublesome points where a = Ωa.

(12) When we got everything we could get out of I, we add more I's to the mix: I2, I3, ... etc.

(13) You can actually get much more juice from the above then you'd might think. Things get complicated quite quickly, before we completely exhaust the power of I's.

(14) When we finally do exhaust it, we introduce M. Now we start the whole process all over again, but on a much much grander scale.

(15) When we exhaust the M's, we use K's.

(16) As for the C function (with it's commas and semicolons), it is nothing more than a convinient short hand to write things I've already talked about.

(17) Finally, a warning: UNOCF does not do things in the accepted way, nor is it even well-defined. If you're serious about learning these topics, you should use a well-defined notation (such as the one in Deedlit's excellent series of posts).

If you use UNOCF, you will learn wrong things which you'll need to spend considerable effort to "unlearn" later.

Any, hope this helps.