This is my attempt to create system, which gives possibility to define fundamental sequences for all limit ordinals up to Large Veblen Ordinal (LVO). Unfortunately I found no one sourse where somebody gives detailed definition for fundamental sequences for all limit ordinals between Gamma_0 and LVO. And that is why I created this FS-system, using only the system of fundamental sequences for binary Veblen function as the prototype. Previously I published part of this system on this page of wikia as well as on my site.
Let , , is a limit ordinal, is first uncountable ordinal.
Fundamental sequences for limit ordinals of the theta-function up to :
where and for
1.3) and ,
1.4) and ,
Note: The theta-function is shown in the two-argument version , if it can be abbreviated as . The theta-function is an extension of the two-argument Veblen function. For countable arguments theta-function is equal to Veblen function and has same fundamental sequences.
Fundamental sequences for limit ordinals of the theta-function up to (as analogy of fundamental sequences for Veblen function):
2.1) and ,
2.2) and ,
2.4) and ,
2.5) and ,
2.9) and .
Fundamental sequences for limit ordinals of the theta-function up to Large Veblen Ordinal :
let and and for then
3.1) and ,
3.2) and ,
3.6) and ,
3.7) and .
Note: if then expressions 3.1-3.5 should rewrite as 1.3-1.7 and expressions 3.6-3.7 should rewrite as 2.1-2.2.
Up to Bachmann-Howard ordinal