User blog comment:B1mb0w/Fundamental Sequences/@comment-5529393-20160124015439/@comment-5529393-20160206060323

Rule 2B does not give proper fundamental sequences because they do not approach the ordinal that they are supposed to. For example, you are giving $$\varphi(2,1)$$ a fundamental sequence of $$\varphi(2,0)\uparrow\uparrow n$$, but that limits to $$\varepsilon_{\varphi(2,0) + 1}$$, not $$\varphi(2,1)$$.

No, I do not agree that values of Veblen functions will vary depending upon the FSes being used. The Veblen functions have a fixed definition, namely

$$\varphi(0,\alpha) = \omega^\alpha$$

For $$\alpha > 0$$, $$\varphi(\alpha, \beta)$$ is the $$\beta$$th ordinal $$\gamma$$ that satisfies $$\varphi(\delta, \gamma) = \gamma$$ for all $$ \delta < \gamma$$.

So your fundamental sequences do not match the official definition of the Veblen functions. I guess what you mean is that you are defining your own Veblen functions, but they are much weaker than the standard versions.