User blog comment:Edwin Shade/How Do I Evaluate BEAF Arrays In Two Dimensions ?/@comment-30754445-20170827183955/@comment-5529393-20170830172853

@PsiCubed2: Good point about Bowers. We could add expressions like (X↑)Xn to the notation, but then we need to know how to reduce something like (X↑)X2. What we want is (X↑)X-1(Xn), but if we start subtracting from X we're going to have infinite regress.

@Edwin Shade: In order for us to answer your questions, you need to define what "unique ordinal infinities" and "fundamentally different" mean. I will say that, if one considers $$\omega$$ and $$\varepsilon_0$$ to be fundamentally different ordinal infinities, then probably one would consider $$\Gamma_0$$, the Bachmann-Howard ordinal, the TFB ordinal, etc to all be different as well, and there would be infinitely many fundamentally different ordinal infinities.

There are fast-growing functions that aren't defined by recursion; see the TREE function for example.

There is nothing "beyond" uncomputable functions, since every function is either computable or uncomputable.