User blog:Bubby3/Analysis of Hyper-Extended Cascading E notation (and further extensions)

We start this with {#,#,1,2} having level phi(1,0,0,0)

({#,#,1,2})#= phi(phi(1,0,0,0),0,1)

({#,#,1,2})##= phi(phi(1,0,0,0),1,0)

({#,#,1,2}){{#,#,1,2}+1}#= phi(phi(1,0,0,0)+1,0,0)

{{#,#,1,2},#,1,2}= phi(1,0,0,1)

{#,#+1,1,2}= phi(1,0,0,w)

&(1)= phi(1,0,0,w^2)

&(2)= phi(1,0,0,w^2*2)

&(#)= phi(1,0,0,w^3)


 * 1) *^#= phi(1,0,1,0)


 * 1) *^##= phi(1,0,1,1)


 * 1) *^#^#= phi(1,0,1,w)


 * 1) *^#*^#= phi(1,0,1,phi(1,0,1,0))


 * 1) *^^#= phi(1,0,2,0)


 * 1) *^^##= phi(1,0,3,0)


 * 1) *^^^#= phi(1,1,0,0)


 * 1) *^^^^#= phi(1,2,0,0)


 * 1) *{#}#= phi(1,w,0,0)

#*{#,#,1,2}#= phi(2,0,0,0)

Beyond this point, it becomes ambigous what comes next, for example how #*{#,#+1,1,2} evualates. Everything beyond this point are just guesses.

#*{#,#+1,1,2}#=phi(2,0,0,w) x^x must reach at least the LVO#x^x/#=theta(W^W*w,0)#x^^#/ reaches the Bachmann-Howard level, where x acts as capital omega So babbulbufihgh has level theta(Ww)
 * 1) *&(1)#=phi(2,0,0,w^2)#**^#=phi(w,0,0,0)
 * 1) /^#=phi(1.0,0,0,0)#x/^#=phi(1,0,0,0,0,0)