Postby **Vytron** » Sat Mar 07, 2015 8:27 am UTC

Okay, here's some ordering.

There's these series (separated by spaces):

0 0,0 0,0,0 0,0,0,0 0,0,0,0,0...

When these series are at the rightmost point in:

n,X

You drop a 0 and return: <(>n<),X>

And then there's these series:

{0} {0},{0} {0},{0},{0} {0},{0},{0},{0} {0},{0},{0},{0},{0}

When these series are at the rightmost point in:

n,X

You transform the rightmost {0} into <,0>

There's:

0,{0} 0,0,{0} 0,0,0,{0} 0,0,0,0,{0} 0,0,0,0,{0}

When these series are at the rightmost point in:

n,X

You drop a 0, and return <,Y> where Y is an early member of the series (0,0,0,0,{0} becomes <0,0,0,{0}>)

When there's ambiguity in the series:

You apply these rules to string a, in n,x,a of the rules of the other post (i.e. x ends in the penultimate biggest element of X).

{0,0} Becomes 0<,0>{0} if it's the rightmost element, otherwise, all previous elements follow the same rules at the right of {0,0}

There exist the series

{0,0},0 {0,0},0,0 {0,0},0,0,0 {0,0},0,0,0,0 {0,0},0,0,0,0,0...

And the series:

{0,0},{0} {0,0},{0},{0} {0,0},{0},{0},{0} {0,0},{0},{0},{0},{0} {0,0},{0},{0},{0},{0},{0}...

And the series:

{0,0},0,{0} {0,0},0,0,{0} {0,0},0,0,0,{0} {0,0},0,0,0,0,{0} {0,0},0,0,0,0,{0}

Which have known reduction rules for {0,0}X

There's the series:

{0,0} {0,0},{0,0} {0,0},{0,0},{0,0} {0,0},{0,0},{0,0},{0,0} {0,0},{0,0},{0,0},{0,0},{0,0}

When these series are at the rightmost point in:

n,X

You transform the rightmost {0,0} into <<,0>,{0}>

There's the series:

0,{0,0} 0,0,{0,0} 0,0,0,{0,0} 0,0,0,0,{0,0} 0,0,0,0,{0,0}

When these series are at the rightmost point in:

n,X

You drop a 0, and return <,Y> where Y is an early member of the series (0,0,0,0,{0,0} becomes <0,0,0,{0,0}>)

When there's ambiguity in the series:

You apply these rules to string a, in n,x,a of the rules of the other post (i.e. x ends in the penultimate biggest element of X).

{0},{0,0} returns <0<,0>{0,0}>

There's the series:

0,{0},{0,0} 0,0,{0},{0,0} 0,0,0,{0},{0,0} 0,0,0,0,{0},{0,0} 0,0,0,0,0,{0},{0,0}

On here you reduce X,{0,0}, create Y=X<,X>{0,0} And return Y<,Y>

I.e.:

0,0,0,{0},{0,0} Y=0,0,{0}<,0,0,{0}>{0,0} = 0,0,{0}<,0,0,{0}>{0,0}<,0,0,{0}<,0,0,{0}>{0,0}>

{0,0,0} returns <<<,0>,{0}>,{0,0}>

{0,0,0,0} returns <<<<,0>,{0}>,{0,0}>,{0,0,0}>

{0,0,0,0,0} returns <<<<<,0>,{0}>,{0,0}>,{0,0,0}>,{0,0,0,0}>

{{0}} returns: {0<,0>}

{{0}} is an element bigger than any other {X} with no {}s on X, so all previous series follow the same rules on {{0}}X.

All other series also follow the same rules on X{{0}} (i.e. to know how {{0}} behaves just pretend {{0}} is a {0,0,0} with one 0 more than any terms in X).

The series:

{{0}} {{0}},{{0}} {{0}},{{0}},{{0}} {{0}},{{0}},{{0}},{{0}} {{0}},{{0}},{{0}},{{0}},{{0}}

If at the rightmost change the rightmost {{0}} to {0<,0>}

The series:

0,{{0}} 0,0,{{0}} 0,0,0,{{0}} 0,0,0,0,{{0}} 0,0,0,0,0,{{0}}

Drop a 0, and return <,Y> where Y is an early member of the series (0,0,0,0,{{0}} becomes <0,0,0,{{0}}>)

Series:

0,{0},{{0}} 0,0,{0},{{0}} 0,0,0,{0},{{0}} 0,0,0,0,{0},{{0}} 0,0,0,0,0,{0},{{0}}

On here you reduce X,{{0}}, create Y=X<,X>{{0}} And return Y<,Y>

0,{0},{0,0},{{0}} 0,0,{0},{0,0},{{0}} 0,0,0,{0},{0,0},{{0}} 0,0,0,0,{0},{0,0},{{0}} 0,0,0,0,0,{0},{0,0},{{0}}

On here you reduce X,{{0}}, create Y=X<,X>{{0}} And return Y<,Y>

{{0},0} returns {0<,0>},{{0}}

That is, to add a 0 to {X}, you have to have the element that created {X} in Y{X}

{{0},0,0} returns {0<,0>},{{0}},{{0},0}

{{0},0,0,0} returns {0<,0>},{{0}},{{0},0},{{0},0,0}

{{0},0,0,0,0} returns {0<,0>},{{0}},{{0},0},{{0},0,0,0}

{{0},{0}} returns {{0}<,0>}

{{0},{0},0} returns {{0}<,0>},{{0},{0}}

{{0},{0},0,0} returns {{0}<,0>},{{0},{0}},{{0},{0},0}

{{0},{0},{0}} returns {{0},{0}<,0>}

And so on.

{0,{0}} returns: {{0}<,{0}>}

{{0,0}} returns: {0<,0>{0}}

{{{0}}} returns: {{0<,0>}}

{0-0} returns: <{>0<}>

{0-0} is an element bigger than any other {X} with no -s on X, so all previous series follow the same rules on {0-0}X.

All other series also follow the same rules on X{0-0} (i.e. to know how {0-0} behaves just pretend {0-0} is a {{{0}}} with one {} nest more than any terms in X).

The series:

{0-0} {0-0},{0-0} {0-0},{0-0},{0-0} {0-0},{0-0},{0-0},{0-0} {0-0},{0-0},{0-0},{0-0},{0-0}

If at the rightmost change the rightmost {0-0} to <{>0<}>

The series:

0,{0-0} 0,0,{0-0} 0,0,0,{0-0} 0,0,0,0,{0-0} 0,0,0,0,0,{0-0}

Drop a 0, and return <,Y> where Y is an early member of the series (0,0,0,0,{0-0} becomes <0,0,0,{0-0}>)

Series:

0,{0},{0-0} 0,0,{0},{0-0} 0,0,0,{0},{0-0} 0,0,0,0,{0},{0-0} 0,0,0,0,0,{0},{0-0}

On here you reduce X,{0-0}, create Y=X<,X>{0-0} And return Y<,Y>

0,{0},{0,0},{0-0} 0,0,{0},{0,0},{0-0} 0,0,0,{0},{0,0},{0-0} 0,0,0,0,{0},{0,0},{0-0} 0,0,0,0,0,{0},{0,0},{0-0}

On here you reduce X,{0-0}, create Y=X<,X>{0-0} And return Y<,Y>

0,{0},{{0}},{0-0} 0,0,{0},{{0}},{0-0} 0,0,0,{0},{{0}},{0-0} 0,0,0,0,{0},{{0}},{0-0} 0,0,0,0,0,{0},{{0}},{0-0}

On here you reduce X,{0-0}, create Y=X<,X>{0-0} And return Y<,Y>

{{0-0}} returns <{>0<}>,{0-0}

{{0-0},0} returns <{>0<}>,{0-0},{{0-0}}

That is, to add a 0 to {X}, you have to have the element that created {X} in Y{X}

{{0-0},0,0} returns <{>0<}>,{0-0},{{0-0},0}

{{0-0},0,0,0} returns <{>0<}>,{0-0},{{0-0},0},{{0-0},0,0}

{{0-0},0,0,0,0} returns <{>0<}>,{0-0},{{0-0},0},{{0-0},0,0},{{0-0},0,0,0}

{{0-0},{0}} returns {{0-0}<,0>}

{{0-0},{0},0} returns {{0-0}<,0>},{{0-0},{0}}

{{0-0},{0},0,0} returns {{0-0}<,0>},{{0-0},{0}},{{0-0},{0},0}

{{0-0},{0},{0}} returns {{0-0},{0}<,0>}

And so on.

{0,{0-0}} returns: {{0-0}<,{0-0}>}

{{{0-0}}} returns: {{<{>0<}>,{0-0}}}

{{{{0-0}}}} returns: {{{<{>0<}>,{0-0}}}}

{0-0,0} returns: <{>0-0<}>

{{0-0,0}} returns: <{>0-0<}>{0-0,0}

{{{0-0,0}}} returns: {<{>0-0<}>{0-0,0}}

{0-0,0,0} returns: <{>0-0,0<}>

{0-{0}} returns: <{>0-0<,0><}>

{0-{0},{0}} returns: <{>0-{0},0<,0><}>

{0-{0-0}} returns: <{>0-<{>0<}><}>

{0-{0-0,0}} returns: <{>0-<{>0-0<}><}>

{0-{0-0,0,0}} returns: <{>0-<{>0-0,0<}><}>

{0-{0-{0}}} returns: <{>0-<{>0-0<,0><}><}>

{0-{0-{0-0}}} returns: <{>0-<{>0-<{>0<}><}><}>

{0-{0-{0-0,0}}} returns: <{>0-<{>0-<{>0-0<}><}><}>

{0-{0-{0-0,0,0}}} returns: <{>0-<{>0-<{>0-0,0<}><}><}>

{0-{0-{0-{0}}}} returns: <{>0-<{>0-<{>0-0<,0><}><}><}>

And:

{0,0-0} = <{0->{0}<}>

Now, please note that {0,0-0} never appeared in 0-X, so we use {0-{0,0-0}} to represent higher values.

{0,0-0} is an element bigger than any other {0-X} with no 0,0- on X, so all previous series follow the same rules on {0,0-0}X.

All other series also follow the same rules on X{0,0-0} (i.e. to know how {0,0-0} behaves just pretend {0,0-0} is a {0-{0-{0-{0}}}} with one - more than any terms in X).

The series:

{0,0-0} {0,0-0},{0,0-0} {0,0-0},{0,0-0},{0,0-0} {0,0-0},{0,0-0},{0,0-0},{0,0-0} {0,0-0},{0,0-0},{0,0-0},{0,0-0},{0,0-0}

If at the rightmost change the rightmost {0,0-0} to <{0->{0}<}>

The series:

0,{0,0-0} 0,0,{0,0-0} 0,0,0,{0,0-0} 0,0,0,0,{0,0-0} 0,0,0,0,0,{0,0-0}

Drop a 0, and return <,Y> where Y is an early member of the series (0,0,0,0,{0,0-0} becomes <0,0,0,{0,0-0}>)

Series:

0,{0},{0,0-0} 0,0,{0},{0,0-0} 0,0,0,{0},{0,0-0} 0,0,0,0,{0},{0,0-0} 0,0,0,0,0,{0},{0,0-0}

On here you reduce X,{0,0-0}, create Y=X<,X>{0,0-0} And return Y<,Y>

0,{0},{0,0},{0,0-0} 0,0,{0},{0,0},{0,0-0} 0,0,0,{0},{0,0},{0,0-0} 0,0,0,0,{0},{0,0},{0,0-0} 0,0,0,0,0,{0},{0,0},{0,0-0}

On here you reduce X,{0,0-0}, create Y=X<,X>{0,0-0} And return Y<,Y>

0,{0},{{0}},{0,0-0} 0,0,{0},{{0}},{0,0-0} 0,0,0,{0},{{0}},{0,0-0} 0,0,0,0,{0},{{0}},{0,0-0} 0,0,0,0,0,{0},{{0}},{0,0-0}

On here you reduce X,{0,0-0}, create Y=X<,X>{0,0-0} And return Y<,Y>

0,{0},{0-0},{0,0-0} 0,0,{0},{0-0},{0,0-0} 0,0,0,{0},{0-0},{0,0-0} 0,0,0,0,{0},{0-0},{0,0-0} 0,0,0,0,0,{0},{0-0},{0,0-0}

On here you reduce X,{0,0-0}, create Y=X<,X>{0,0-0} And return Y<,Y>

{{0,0-0}} returns <{0->{0}<}>,{0,0-0}

{{0,0-0},0} returns <{0->{0}<}>,{0,0-0},{{0,0-0}}

That is, to add a 0 to {X}, you have to have the element that created {X} in Y{X}

{{0,0-0},0,0} returns <{0->{0}<}>,{0,0-0},{{0,0-0}},{{0,0-0},0}

{{0,0-0},0,0,0} returns <{0->{0}<}>,{0,0-0},{{0,0-0}},{{0,0-0},0},{{0-0},0,0}

{{0,0-0},0,0,0,0} returns <{0->{0}<}>,{0,0-0},{{0,0-0}},{{0,0-0},0},{{0-0},0,0},{{0-0},0,0,0}

{{0,0-0},{0}} returns {{0,0-0}<,0>}

{{0,0-0},{0},0} returns {{0,0-0}<,0>},{{0,0-0},{0}}

{{0,0-0},{0},0,0} returns {{0,0-0}<,0>},{{0,0-0},{0}},{{0,0-0},{0},0}

{{0,0-0},{0},{0}} returns {{0,0-0},{0}<,0>}

And so on.

{0,{0,0-0}} returns: {{0,0-0}<,{0,0-0}>}

{{{0,0-0}}} returns: {<{0->{0}<}>,{0,0-0}}

{{{{0,0-0}}}} returns: {{{<{0->{0}<}>,{0,0-0}}}}

{0-{0,0-0}} returns: <{>0,0-0<}>

So:

3{0-{0,0-0}} = {{{{0,0-0}}}} = {{{{0-{0-{0-{0-{0}},{0,0-0}}}}

...

So you were right, these don't match what I've said in previous posts. But still, {0-{0,0-0}} dominates everything that doesn't have a "0-{0,0-" somewhere, and 3{0-{0,0-0}} should go past f_φ(2,φ(3,0)+1)(3) in the FGH.

There's also these series:

{0-{0,0-0}} {0-{0-{0,0-0}}} {0-{0-{0-{0,0-0}}}}, {0-{0-{0-{0-{0,0-0}}}}}

Which dominate anything without a "0-{0,0-", "0-{0-{0,0-" "0-{0-{0-{0,0-"...

And:

{0,0-0,0} = <{0->0,0-0<}>

Which has to go again:

{0-{0,0-0,0}}} {0-{0-{0,0-0,0}}}} {0-{0-{0-{0,0-0,0}}}}}, {0-{0-{0-{0-{0,0-0,0}}}}}}

{0,0-0,0,0}}} = <{0->{0,0-0,0}<}>

And:

{0,0-{0}} = <{0->{0,0-0<,0>}<}>

And once you run out of things to do you add a 0- so in {0-0-0} you have to nest bigger values to reach {0-0-0,0}.