Yeh, I dont’ wanna touch that yet. I’d rather use yours.Moosey wrote: ↑March 3rd, 2020, 9:36 amActually this use of sup is questionabletestitemqlstudop wrote: ↑March 2nd, 2020, 6:54 amE(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}
Ordinals in googology
Re: Ordinals in googology
Re: Ordinals in googology
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
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
Last edited by Moosey on March 4th, 2020, 10:11 am, edited 1 time in total.
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Re: Ordinals in googology
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
∃(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
Re: Ordinals in googology
This can be simplified a bittestitemqlstudop wrote: ↑March 2nd, 2020, 6:54 amE(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)-$)-(x-1), etc}
E(0) = 0
E($-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-$)-x, etc}
This makes it considerably more legible, of course at the price of trees- (0-0) is the same tree as 0-0, but of course it's still a notation
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Re: Ordinals in googology
As I'm incredibly bored, I think I'll describe an ordinal-inspired array system like TAN.
So
a is a number
a{} = a
a{$,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
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{$,b,^(c-1)!b,^(c-1)!b,...} with d bs
w of course becomes a, but only when it clogs the system
In fact
Let's call this separator notation, SeN
In SeN, let % be a generic separator
%^c = %%%%%% with c %
! is a separator needed for generic separation when , is not appropriate
Brackets are used if necessary to show what is being applied to by separator ops
So for instance
3{w,^w!w} is 3{w,,,w} which is 3{w,,w,,w} which is, as stated, VERY large
now
3{w,^w!w,^w!w} = 3{w,,,,,....,,,,,,,,w} with about 3{w,,w,,w} ,s
So now let's have more fun
Since w "clogs the system" only when left alone, and we can subtract one from w+1, which is technically not disallowed, ...
3{w,^(w+1)!w} = that
3{w,^(w+2)!w} = about that many w,^w!s
Next
%^^ is shorthand for [%^]^
So
a,[^]^m+1!n+1!b = a,[^]^m!b{a,[^]^m+1!^n!b}!a
a,[^]^1!b!c = a,^b!c
3{w,[^]^w!w!w} is big
not active here but active on discord
Re: Ordinals in googology
I challenge someone to analyse my notation (ASEN, or Applied SEparator Notation) above
a{} = a
a{$,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
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
not active here but active on discord
Re: Ordinals in googology
Without w
3{3,3} = 363
Without ,,
3{w,w} = 366
Without !
3{w,,w} = a super-absurdly large number
Ok that got large quick
3{3,3} = 363
Code: Select all
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
3{w,w} = 366
Code: Select all
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
3{w,,w} = a super-absurdly large number
Code: Select all
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
Re: Ordinals in googology
If this is accidentally posted, do not reply until it is deleted.
Without [^]^
3{w,^w!w} =
Without w+n (as requested by Moosey)
3{w,[^]^w!w!w} = add some curly brackets please i cannot comprehend it
Without [^]^
3{w,^w!w} =
Code: Select all
3{w,^w!w} = 3{w,^3!3} = 3{w,^2!w,^2!w,^2 does this end with } or !w}
3{w,[^]^w!w!w} = add some curly brackets please i cannot comprehend it
Re: Ordinals in googology
screw the above post
I want to share this REAL bad
So, googology discord user Despacit2.0 and I conceived "ARE notation"
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?
I want to share this REAL bad
So, googology discord user Despacit2.0 and I conceived "ARE notation"
Code: Select all
(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)
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?
not active here but active on discord
Re: Ordinals in googology
bump
update: we're at phi(4,0)
hopefully we'll get to phi(w,0) and then g0 soon
update: we're at phi(4,0)
hopefully we'll get to phi(w,0) and then g0 soon
not active here but active on discord
Re: Ordinals in googology
bump
Since test disappeared from the forums this thread has been dead lol
Since test disappeared from the forums this thread has been dead lol
not active here but active on discord