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The last post was not good; but, at least, it teached me to clear everything up before going forward (and other users told me that too :P).

You can check Wythagoras' definition here!

Analysis 1

The analysis is at Wythagoras' blog post.

Extension 1

f_{\#,\alpha+1}(\beta)=f_{\#,\alpha}^\beta(\beta)

If \alpha is a successor ordinal,

f_{\#,\alpha,0,...,0}(\beta)=\underbrace{f_{\#,\alpha-1,f_{\#,\alpha-1,f_{\#,\alpha-1,...f_{\#,\alpha-1,0,...,0}(\beta),...,...,0}(\beta),...,0}(\beta),...,0}(\beta),...,0}_\beta

If \alpha is a limit ordinal,

f_{\#,\alpha,0,...,0}(\beta)=lim_{n\mapsto\omega}f_{\#,\alpha[n],0,...,0}(\beta)

f_{0,\#}(\alpha)=f_\#(\alpha)

Analysis 2

f_{1,0}(\omega)=f_{0,f_{0,f_{0,...f_{0,f_{0,0}(\omega)}(\omega)...}(\omega)}(\omega)}(\omega)=f_{f_{f_{...f_{\omega+1}(\omega)...}(\omega)}(\omega)}(\omega)=\theta(\Omega_\omega)

f_{1,0}(f_{1,0}(\omega))=\theta(\Omega_{\theta(\Omega_\omega)})

f_{1,1}(\omega)=f_{1,0}(f_{1,0}(f_{1,0}(...(f_{1,0}(\omega)...)))=\theta(\Omega_\Omega)

Darn. I'm extremely confused about this.

THIS WORK IS NOT IN PROGRESS ANYMORE BECAUSE OF EXTREME CONFUSION.

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