User blog comment:Edwin Shade/How do you evaluate extended Veblen notation ?/@comment-5529393-20171025083611

It looks like you are attempting to come up with fundamental sequences for the extended Veblen function yourself, so I'll let you have at it. I'll just comment on the Schutte Klammersymbolen.

Basically, the Klammersymbolen is a notation that extends the Extended Veblen function to transfinitely many arguments. The one problem: we can't actually write out infinitely many arguments! The answer is to only write down the arguments that are nonzero, of which there can only be finitely many. But, then we can't tell which argument it is just by where it is located in the function input. So, we have to write down both the value of the argument and which argument it is.

Example: instead of phi(2,5,0,0,3,0,7,2,0), we can write phi(2@8, 5@7, 3@4, 7@2, 2@1), since we have a 2 in the 8th argument (counting from the right, starting at 0 of course), a 5 in the 7th argument, a 3 in the 4th argument, etc. The advantage of the latter way of writing things is that we can now write something like phi(w@epsilon_1, 3@w+1, w^2@w, 5@3) - that is, we can now write infinite ordinals for the places.

So that's good, but now we need to define the notation for infinite places.

The basic definition is: phi(a_1@b_1, ..., a_n@b_n, c@0) is defined to be the cth ordinal that satisfies all equations of the form: phi(a_1@b_1,..., d@b_n, e@f) = e for d < a_n, f < b_n.

Exercise: Show that this definition gives the correct definition for the Veblen function when the place values can only be 0 or 1, and that it gives the correct definition of the Extended Veblen function when the place values must be finite.

For example: Take phi(1@w), or phi(1@w, 0@0). By the definition, this is the smallest ordinal that satisfies all equations of the form phi(d@w, e@f) = e for d < 1, f < w. But that means d must be 0, so we get phi(0@w, e@f) = phi(e@f) = e for all f < w.  That is, we want the smallest ordinal that is the solution to f(e) = phi(e@f) = e for all finite f, i.e. we want phi(e,0,...,0) = e for any number of 0's.  This happens to be the Small Veblen Ordinal. Similarly, phi(1@w, c@0) is the cth ordinal that satisfies phi(e@f) = e for f finite.

Another example: phi(2@w^2, w@w+1, c@0). By the definition, this is the cth ordinal that satisfies all equations of the form phi(2@w^2, d@w+1, e@f) = e for all d < w, f < w+1. Now, phi(2@w^2, d@w+1, e@f) is an increasing function with respect to f, so if the equation is satisfied for larger f, it is satisfied for smaller f. The largest value f can take is w.  So if e satisfies all the equations where f = w, then it will satisfy all the equations where f < w as well. So we need only concern ourselves with the equations of the form phi(2@w^2, d@w+1, e@w) = e for d < w. The cth ordinal that satisfies this equation for all finite d is our ordinal.

Hope this helps! Feel free to ask if you still don't understand.