User:Hyp cos/Catching Function Analysis p2

Go to page 1,2 =Catching Hierarchy I=

From \(C(\Omega^2)\) to \(C(\Omega^\Omega)\)
From \(C(\Omega^2)\) to \(C(\Omega^3)\)

Get \(C(\Omega^2+\Omega\psi(\psi_{I(1,\omega)}(1)))\)

From \(C(\Omega^2+\Omega\psi(\psi_{I(1,\omega)}(1)))\) to \(C(\Omega^22)\)

From \(C(\Omega^22)\) to \(C(\Omega^3)\)

This block is similar to from \(C(\Omega2)\) to \(C(\Omega^2)\) block.

From \(C(\Omega^3)\) to \(C(\Omega^{C(\Omega)})\)

From \(C(\Omega^3)\) to \(C(\Omega^\omega)\)

Very simple here.

From \(C(\Omega^\omega)\) to \(C(\Omega^{\omega+1})\)

Let [*] donates separator {{0{0,1*}1}*}, but just here.

Here \(\psi_{I(I(1,0,0),1)}(0)\) is the supermum of \(\psi_{I(\alpha,I(1,0,0)+1)}(0)\) for all \(\alpha<I(1,0,0)\). Then \(\psi_{I(I(1,0,0),1)}(1)\) is the supermum of \(\psi_{I(\alpha,I(1,0,0)+1)}(\psi_{I(I(1,0,0),1)}(0))\) for all \(\alpha<I(1,0,0)\). Then \(\psi_{I(I(1,0,0),2)}(0)\) is the supermum of \(\psi_{I(\alpha,I(I(1,0,0),1)+1)}(0)\) for all \(\alpha<I(1,0,0)\).