Ordinals in googology

[PkmnQ]

Actually this use of sup is questionable

E(0) = 0
E($-0) = $ + 1, if $ is not in parentheses
E((0-a)-0) = sup{a,a-a,a-a-a,a-a-a-a,etc}
E(x) = limit of E if all nodes are <= x-1
E($-x) = E(($-(x-1))-($-(x-1))-(x-1)) + 1, if $ not in parentheses
E((x-$)-x) = sup{((x-1)-$)-(x-1), ((x-1)-$)-((x-1)-$)-(x-1), ((x-1)-$)-((x-1)-$)-((x-1)-$)-(x-1), etc}

Yeh, I dont’ wanna touch that yet. I’d rather use yours.

[Moosey]

Thought of the day
Username5243 mentions a possibility for Uncountable FSes in UNOCF. (Note: UNOCF is known for being illdefined/hated).

So that means
A transfinite HH

In a nutshell

f_a(b) is defined (or undefined) thusly:

f_0(b) = b
f_(a+1)(b) = f_a(b+1)
f_a(b) = f_a[{b}](b), when a is a limit ordinal with cof >= b
f_a(b) = b, b > cof(a)

So now
What is [{}] ?
Well it's an extension to FSes obviously
Basically
a[{b}] is defined if cof(a) > b
a[{b}] is like a[ b] (space to avoid bbcode) except you apply all sorts of FS rules to all regulars (Insert unformalized frustration here)

Ex
Ω2[{w+3}] = Ω+Ω[{w+3}] = Ω+w+3

Ω_7[{Ω_6}] = Ω_6

I can't formalize it unfortunately.

But for instance
f_W(w+6) = f_w+6(w+6) = f_w(w+12) = w+12
f_I(M) = M
f_W+6(w) = f_W(w+6) = w+12
etc

Probably could be made more useful, or more interesting at least, if someone found a better version of the b > cof(a) rule

[PkmnQ]

I just realized the original definition of E($) uses trees with only zeroes. So maybe we could shorten the notation. Meanwhile, I'll just use the zeroes notation.

∃(0) = 0
∃(ø) = (0-∃(ø[0]))-0
∃(x+1) = ∃(x)-0

I’ll test it.
∃(w) = (0-∃(0))-0 = (0-0)-0
E((0-0)-0) = sup{0,0-0,0-0-0,...} = sup{0,1,2,3,...} = w
∃(w^2+1) = ∃(w^2)-0 = (0-∃(w))-0-0 = (0-(0-0)-0)-0-0
E((0-(0-0)-0)-0-0) = E((0-(0-0)-0)-0) + 1 = sup{(0-0)-0,none of this makes sense}

Janskchdhwhjshsjssahjhsjchcjchchchhhd I do not want to touch the big one

[Moosey]

E(0) = 0
E($-0) = $ + 1, if $ is not in parentheses
E((0-a)-0) = sup{a,a-a,a-a-a,a-a-a-a,etc}
E(x) = limit of E if all nodes are <= x-1
E($-x) = E(($-(x-1))-($-(x-1))-(x-1)) + 1, if $ not in parentheses
E((x-$)-x) = sup{((x-1)-$)-(x-1), ((x-1)-$)-((x-1)-$)-(x-1), ((x-1)-$)-((x-1)-$)-((x-1)-$)-(x-1), etc}

This can be simplified a bit
E(0) = 0
E($-0) = $ + 1
E((0-a)) = sup{a,a-a,a-a-a,a-a-a-a,etc}
E(x) = limit of E if all nodes are < x
E($-(x+1)) = E(($-x)-($-x)-x) + 1, if $ not in parentheses
E(((x+1)-$)-(x+1)) = sup{(x-$)-x, (x-$)-(x-$)-x, (x-$)-(x-$)-(x-$)-x, etc}

This makes it considerably more legible, of course at the price of trees- (0-0) is the same tree as 0-0, but of course it's still a notation

[Moosey]

As I'm incredibly bored, I think I'll describe an ordinal-inspired array system like TAN.

So
a is a number
a{} = a
a{$,0} = a{$}
a{b} = a+b
a{$,b,c+1} = (a+1){$,(a+1){$,(a+1){$,(a+1){...(a+1){$,b,c}...},c},c},c}) with a copies of a+1
a{$,w} = a{$,a}

So for instance
2{2,1} = 3{3{2}} = 3^9
Not much, okay
2{2,2} = 3{3{2,1},1} = 3{4{4{4{2}}},1} = 3{4^4^16,1} = 4^4^4^4^4^16
Still not much
What about
2{w,w}?
Well...
3{3{w,1},1} = 3{4{4{4{4}}},1}
Still not very good, okay

But that's only the first part!
Now for this part
a{$,b,,c} = a{$,b,b,b,...b} w/ c bs
3{w,,w} = 3{w,w,w} = 3{w,w,3} = a whole darn lot
But what's much bigger
Is 3{w,,w,,w}
Which is ultimately: 3{w,w,w,w,w,w,w.........with lots of ws...w,w,w,w}

Of course
That means we can generalize like with ExE (come to think of it this is like ExE)
meaning
a{$,b,^c!d} = a{$,b,^(c-1)!b,^(c-1)!b,...} with d bs
w of course becomes a, but only when it clogs the system

In fact
Let's call this separator notation, SeN
In SeN, let % be a generic separator
%^c = %%%%%% with c %
! is a separator needed for generic separation when , is not appropriate
Brackets are used if necessary to show what is being applied to by separator ops

So for instance
3{w,^w!w} is 3{w,,,w} which is 3{w,,w,,w} which is, as stated, VERY large

now
3{w,^w!w,^w!w} = 3{w,,,,,....,,,,,,,,w} with about 3{w,,w,,w} ,s

So now let's have more fun
Since w "clogs the system" only when left alone, and we can subtract one from w+1, which is technically not disallowed, ...
3{w,^(w+1)!w} = that
3{w,^(w+2)!w} = about that many w,^w!s

Next
%^^ is shorthand for [%^]^

So
a,[^]^m+1!n+1!b = a,[^]^m!b{a,[^]^m+1!^n!b}!a
a,[^]^1!b!c = a,^b!c

3{w,[^]^w!w!w} is big

[Moosey]

I challenge someone to analyse my notation (ASEN, or Applied SEparator Notation) above

a{} = a
a{$,0} = a{$}
a{b} = a+b
a{$,b,c+1} = (a+1){$,(a+1){$,(a+1){$,(a+1){...(a+1){$,b,c}...},c},c},c}) with a copies of a+1
a{$,w} = a{$,a}
a{$,b,,c} = a{$,b,b,b,...b} w/ c bs
a{$,b,^c!d} = a{$,b,^(c-1)!b,^(c-1)!b,...} with d bs
a,[^]^m+1!n+1!b = a,[^]^m!b{a,[^]^m+1!^n!b}!a
a,[^]^1!b!c = a,^b!c

In SeN, let % be a generic separator
%^c = %%%%%% with c %
! is a separator needed for generic separation when , is not appropriate
Brackets are used if necessary to show what is being applied to by separator ops

[PkmnQ]

Without w

3{3,3} = 363

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3{3,3} = 4{4{4{3,2},2},2} = 4{4{123,2},2} = 4{243,2} = 363
4{3,2} = 5{5{5{5{3,1},1},1},1} = 5{5{5{33,1},1},1} = 5{5{63,1},1} = 5{93,1} = 123
5{3,1} = 6{6{6{6{6{3,0},0},0},0},0} = 6{6{6{6{6{3}}}}} = 33
5{33,1} = 6{6{6{6{6{33,0},0},0},0},0} = 6{6{6{6{6{33}}}}} = 63
4{123,2} = 5{5{5{5{123,1},1},1},1} = 5{5{5{153,1},1},1} = 5{5{183,1},1} = 5{213,1} = 243

Without ,,

3{w,w} = 366

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3{w,w} = 3{w,3} = 4{4{4{w,2},2},2} = 4{4{126,2},2} = 4{246,2} = 366
4{w,2} = 5{5{5{5{w,1},1},1},1} = 5{5{5{36,1},1},1} = 5{5{66,1},1} = 5{96,1} = 126
5{w,1} = 6{6{6{6{6{w,0},0},0},0},0} = 6{6{6{6{6{w}}}}} = 6{6{6{6{6{6}}}}} = 36

Without !

3{w,,w} = a super-absurdly large number

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3{w,,w} = 3{w,,3} = 3{w,w,w} = 3{w,w,3} = 4{w,4{w,4{w,w,2},2},2}
4{w,w,2} = 5{w,5{w,5{w,5{w,w,1},1},1},1}
5{w,w,1} = 6{w,6{w,6{w,6{w,6{w,w,0},0},0},0},0} = 6{w,6{w,6{w,6{w,6{w,w}}}}} = 6{w,6{w,6{w,6{w,6{w,6}}}}} = 6{w,6{w,6{w,6{w,3991692}}}} = an absurdly large number
6{w,6} = 7{7{7{7{7{7{w,5},5},5},5},5},5} = 3991692
7{w,5} = 8{8{8{8{8{8{8{w,4},4},4},4},4},4},4} = 665292
8{w,4} = 9{9{9{9{9{9{9{9{w,3},3},3},3},3},3},3},3} = 9{9{9{9{9{9{9{11892,3},3},3},3},3},3},3} = 11880*7+11892
9{w,3} = 10{10{10{10{10{10{10{10{10{w,2},2},2},2},2},2},2},2},2} = 95052
10{10{10{10{10{10{10{10{1332,2},2},2},2},2},2},2},2} = 10{10{10{10{10{10{10{2652,2},2},2},2},2},2},2} = 11892
10{w,2} = 11{11{11{11{11{11{11{11{11{11{w,1},1},1},1},1},1},1},1},1},1} = 11{11{11{11{11{11{11{11{11{144,1},1},1},1},1},1},1},1},1} = 11{11{11{11{11{11{11{11{276,1},1},1},1},1},1},1},1} = 11{11{11{11{11{11{11{408,1},1},1},1},1},1},1} = 1332
11{w,1} = 12{12{12{12{12{12{12{12{12{12{12{w}}}}}}}}}}} = 12{12{12{12{12{12{12{12{12{12{12{12}}}}}}}}}}} = 144
11{144,1} = 12{12{12{12{12{12{12{12{12{12{12{144}}}}}}}}}}} = 276

Ok that got large quick

[Moosey]

Do 3{w[^]^w!w!w}

[PkmnQ]

If this is accidentally posted, do not reply until it is deleted.

Without [^]^

3{w,^w!w} =

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3{w,^w!w} = 3{w,^3!3} = 3{w,^2!w,^2!w,^2 does this end with } or !w}

Without w+n (as requested by Moosey)

3{w,[^]^w!w!w} = add some curly brackets please i cannot comprehend it

[Moosey]

screw the above post
I want to share this REAL bad


So, googology discord user Despacit2.0 and I conceived "ARE notation"

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(x)A = x + 1
(x)R = ((...(x)x)x)...
(x)(y)(z)(.....)(a)(b)R = ((x)(y)(z)(.....)(((...(a)...b)b)b)
(...)[#]E = (...)(#########... (the last parenthesis pair is deleted)

And we've made an analysis sheet
but we've gotten stuck at zw = sup{(A)[(R)(E)]E, (A)[([(R)(E)]E)(E)]E, (A)[([([(R)(E)]E)(E)]E)(E)]E...}
does anyone have an idea for an extension?

[Moosey]

bump
update: we're at phi(4,0)
hopefully we'll get to phi(w,0) and then g0 soon

[Moosey]

bump
Since test disappeared from the forums this thread has been dead lol